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It can only happen if the body has chaotic rotation:

Chaotic rotation involves the irregular and unpredictable rotation of an astronomical body. Unlike Earth's rotation, a chaotic rotation may not have a fixed axis or period. Because of the conservation of angular momentum, chaotic rotation is not seen in objects that are spherically symmetric or well isolated from gravitational interaction, but is the result of the interactions within a system of orbiting bodies, similar to those associated with orbital resonance.

Examples of chaotic rotation include Hyperion, a moon of Saturn, which rotates so unpredictably that the Cassini probe could not be reliably scheduled to pass by unexplored regions, and Pluto's Nix, Hydra, and possibly Styx and Kerberos, and also Neptune's Nereid.

Another example is that of galaxies; from careful observation by the Keck and Hubble telescopes of hundreds of galaxies, a trend was discovered that suggests galaxies such as our own Milky Way used to have a very chaotic rotation, with planetary bodies and stars rotating randomly.

However note that, since chaotic rotation is not seen in objects that are spherically symmetric, it cannot happen on planets, which are, by definition, under hydrostatic equilibrium and thus spherically symmetric.

You need a new source of "night".

Start with a normal spherical planet with a consistent rotation speed and a subsequently predictable 30 hour day/night patterns, then add dense high-atmospheric clouds which block all sunlight from reaching the surface.

Then all you need is erratic high-atmosphere wind patterns, so that surface dwellers can never predict when their daylight will be blocked by the clouds.

This works best if there is some cohesion among the particles which make up the clouds makes them clump together in continent spanning masses. Smaller clumps might cause a lot of midday eclipses but wouldn't significantly effect the length of productive daylight. But when one of the big ones floated in over your city, there is no telling how long it will be before you see daylight again.

All of this could be the result of an apocalyptic eco-war where nanites were released into the upper atmosphere to destroy the enemy's agriculture, or it could be natural, the result of a recent super volcano or even the planet passing through a river of space dust.

Simulated reality.

Very advanced technology in the background, i.e. the planet used to be very high tech, has orbital mirrors which control where the sunlight goes, and they've regressed.

A very shiny moon would probably come close to this although that depends on how low a bar you will allow for "daylight".

Now if you mean "our level of technology with anything close to normal orbital realities" the answer is "no".

No, this is not possible.

The length of day is determined by the rotation period of the planet around its own axis. The rotation speed would have to change dramatically to accomplish different day-lengths.

What could be similar is a strange arrangement of twin planets, where the rotation axis of the two planets spinning around each other is not at 90° of the solar plane, so the day length would change throughout the year for certain locations. Otherwise, since these two planets would be tidally locked, the rotation period of one planet (one planet day, so to speak) would be the same as the rotation period of the two planets around each other.

And yet it could work!

Our daytime is always the same because both the rotation and translation movements of the Earth are periodic.

However, you can have nonperiodic trajectories for a planet under the influence of a two-star system, as described in the three-body problem.

Under certain initial conditions these trajectories can be repetitive but could get "chaotic", leading to different periods of light and darkness

Without physics or computers, daytimes would seem chaotic to the inhabitants of such a planet. In fact, under such conditions it is hard to think that a civilization with complex math could emerge.

Check out these links:

https://www.wolframscience.com/nks/notes-7-4--three-body-problem/

https://medium.com/@mikeharrisNY/misconceptions-about-the-three-body-problem-and-its-relation-to-forecasting-c0c0a2bf44cc

Various plans have been floated in various places to modulate a planet's sunlight, either by adding shades to reduce it (to make Venus less toasty, or perhaps to offset our own global warming) or adding mirrors and lenses to amplify it (to make Mars more toasty; KSR's Soletta in his Mars trilogy is a good example).

Positioning such devices and keeping them on station isn't going to be easy... they're massive, and the pressure of sunlight has to be offset somehow. They're also quite fragile. A few damaged, off-station or otherwise out-of-control sunlight modifiers might conceivably create twilight or eclipses at intervals that may be or become predictable given enough time and brainpower brought to bear, but for some period of time they could easily seem quite random.

A wobbling rotation axis is will do.

On Earth, length of day changes through the seasons fue to the axial tilt. We don't really notice it on low latitudes. But on higher latitudes a summer day can last from 13 hours to six months, depending on where you are.

Now, the Earth also wobbles, though very little:

The Chandler wobble or variation of latitude is a small deviation in the Earth's axis of rotation relative to the solid earth, which was discovered by American astronomer Seth Carlo Chandler in 1891. It amounts to change of about 9 metres (30 ft) in the point at which the axis intersects the Earth's surface and has a period of 433 days.

(...)

The Chandler wobble is an example of the kind of motion that can occur for a spinning object that is not a sphere; this is called a free nutation. Somewhat confusingly, the direction of the Earth's spin axis relative to the stars also varies with different periods, and these motions—caused by the tidal forces of the Moon and Sun—are also called nutations, except for the slowest, which are precessions of the equinoxes.

(...)

One hypothesis for the source of the wobble was proposed in 2001 by Richard Gross at the Jet Propulsion Laboratory managed by the California Institute of Technology. He used angular momentum models of the atmosphere and the oceans in computer simulations to show that from 1985 to 1996, the Chandler wobble was excited by a combination of atmospheric and oceanic processes, with the dominant excitation mechanism being ocean‐bottom pressure fluctuations. Gross found that two-thirds of the "wobble" was caused by fluctuating pressure on the seabed, which, in turn, is caused by changes in the circulation of the oceans caused by variations in temperature, salinity, and wind. The remaining third is due to atmospheric fluctuations.

Now, if we amped up Chandler's wobble to be of thousands of kilometers rather than just nine, this would cause days to change length much faster. This would combine with seasons for some really conplicated cycles.

The fun part is that wobbles combine:

This wobble [Chandler's] which is a nutation, combines with another wobble with a period of one year, so that the total polar motion varies with a period of about 7 years.

Combined wobbles make cycles much harder to calculate.

So just give a few wobbles to your planet, caused by a variety of sources, and you're set.

The short answer is no, this can't be done for a planet. While you could have a planet orbiting 2 or more stars, that planet would be savagely inhospitable to life as we know it. If you had a life form that could handle temperature changes of hundreds of degrees Celsius over the course of one full orbit then maybe, but if you want a planet with a reasonably consistent environment then you need a stable, reasonably circular orbit around a single star.

You can get variability in the day/night cycle from a variety of things, but all of them are completely predictable. Two options from our own experience:

• Axial tilt - produces seasonal variations just like here on Earth.


• Eclipsing bodies - other objects (like our moon) can block all or part of the illumination from the star.

Also in our solar system but away from Earth, there are planet-sized moons orbiting Jupiter which regularly pass through Jupiter's shadow. This is also completely predictable and regular as clockwork.

So assuming that you want your planet to be habitable and stable you're not going to be able to find a natural method for changing the day/night cycle that isn't completely predictable. Nothing that fits with the science of our universe at least.

Given that, it's clear that any environment that has a random element to its day/night cycle is artificial. It might take the inhabitants a while to twig to this, if they ever manage to get past the inevitable problem of trying to explain the universe with such obviously bad information.

Cixin Liu, in "The Three Body Problem" describes evolution on a planet in a trinary star system. The orbits of three stars in relation to each other is so complex that it seems random and civilisations collapse numerous times when caught by an unexpected freezing or heating cycle. The Three body problem is a scientific thing (Wikipedia) and is generalised as the "N-body problem".

Cixin Liu's book is well worth a read and seems entirely relevant to the scenario you are exploring.

One possibly way would be to have your planet orbit very close to a very dim red dwarf star. So close that it would normally be tidally locked to the star with one side always facing the star and one side always facing away from the star.

But instead the "planet" is actually a giant, Earth-sized natural satellite or moon of a gas giant planet that orbits in the habitable zone of the star. So instead of being tidally locked to the star it will be tidally locked to the gas giant and the star will rise and set on the moon so the moon will have days and nights instead of eternal day in one hemisphere and eternal night in the other hemisphere.

Now change the axial tilt of the planet, and thus of its moon, so that it is nearly 90 degrees, and thus the axis of the planet and of the orbiting moon will be almost pointed in the plane of the planet's orbit around the star.

That means that at one time, the north poles of the planet and the moon will be pointing almost directly at the star, and the northern hemisphere of the moon will have constant day and the southern hemisphere will have constant night.

And when the planet moves 90 degrees along its orbit around the star, the direction to the star will now be at a 90 degree right angle to the direction of the planet and moon's axis, and both hemispheres of the moon will have normal days and nights based on the period of the moon's orbit around the planet.

And when the planet moves 90 more degrees along its orbit around the star, a total of 180 degrees from the first point, the south poles of the planet and the moon will be pointing almost directly at the star, and the southern hemisphere of the moon will have constant day and the northern hemisphere will have constant night.

And when the planet moves 90 more degrees along its orbit around the star, a total of 270 degrees from the first point, the direction to the star will now be at a 90 degree right angle to the direction of the planet and moon's axis, and both hemispheres of the moon will have normal days and nights based on the period of the moon's orbit around the planet.

And when the planet moves 90 more degrees along its orbit around the star, a total of 360 degrees from the first point, it will be back at the first point, the north poles of the planet and the moon will be pointing almost directly at the star, and the northern hemisphere of the moon will have constant day and the southern hemisphere will have constant night.

And at intermediae positions along their orbit the moon will have days and nights of varying length between those extremes.

And what will be the relative length of the moon's orbit around its gas giant planet and their common orbit around their star?

A giant, Earth-sized habitable moon orbiting around a gas giant should have an orbital period lasting about 1.5 to 15 Earth days - possibly longer or shorter.

A gas giant planet and its habitable moon could have a orbital period around a tiny red giant star of only about 5 Earth days and still be within the habitable zone of the star. The four planets of star TRAPPIST-1 that orbit within its habitable zone have orbital periods, or years, of 12.4, 9.2, 6.1, and 4.05 Earth days.

Thus at first sight it would seem possible for the "day" of the moon to be several times as long as its year.

The article "Exomoon Habitability Constrained by Illumination and Tidal heating" by Rene Heller and Roy Barnes Astrobiology, January 2013, discusses factors affecting the habitability of exomoons.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3549631/

It states that for the moon to have a stable orbit, the orbital period of its planet around the star has to be at least nine times as long as the orbital period of the moon around the planet.

So if it takes about 1.5 to 15 Earth days for the moon to orbit it's planet, it should take the planet at least about 13.5 to 135 Earth days for the planet to orbit their star, in order for the moon to have a stable orbit.

So assume that the orbit of the moon around its planet last for exactly 2.0 earth days and the orbit of the planet around it's star lasts for exactly 16 moon days, or 32.0 Earth days.

So when the moon and planet are in the first position mentioned above, the northern hemisphere will be in a day that lasts for several days of the moon, and the southern hemisphere will be in a night that lasts for several days of the moon.

Eight Earth days, and four moon orbits later, the planet and moon will be in the second position 90 degrees along their orbit mentioned above, and the moon will have a day-night cycle like Earth's, but the days and nights will be twice as long as Earth days and nights.

Sixteen Earth days, and eight moon orbits after the first position, the planet and moon will be in the third position 180 degrees along their orbit mentioned above, the southern hemisphere will be in a day that lasts for several days of the moon, and the northern hemisphere will be in a night that lasts for several days of the moon.

Twenty four Earth days, and twelve moon orbits after the first position, the planet and moon will be in the fourth position 270 degrees along their orbit mentioned above, and the moon will have a day-night cycle like Earth's, but the days and nights will be twice as long as Earth days and nights.

Thirty two Earth days, and sixteen orbits after the first position, the planet and moon will be in the fifth position 360 degrees along their orbit mentioned above, and thus back in the first position, and the northern hemisphere will be in a day that lasts for several days of the moon, and the southern hemisphere will be in a night that lasts for several days of the moon.

And in intermediate positions along the orbit the lengths of days and nights will vary between the above extremes.

Also there will be positions in their orbits where and when light reflected off the gas giant planet may provide considerable illumination to the moon, and there may be positions in their orbits where and when the gas giant planet eclipses the star as seen from the moon, giving the moon an eclipse longer than eclipses on Earth, or a much shorter night than usual.

Of course the example I gave, when the orbital period of the moon was exactly two Earth days long, and the orbital period of the planet around the star was exactly as long as sixteen of the moon's orbital period around the planet, was highly oversimplified. It would be an incredibly unlikely coincidence for one orbital period to be an exact multiple of the other.

And you may be able to use a program to design your star system and try out various arrangements of your moon, planet, and star to see which gives the best variation in day length.

If you want to complicate the cycles of light and dark on your world more, you can add another star to the star system, giving the moon another source of light and maybe of heat as well.

The distance between the star that the planet and the moon orbit and the other star should be at least five times the distance between the first star and the planet and the moon.

So the other star should be at least five times as far away from the planet and moon as the star that they orbit. And if the second star has the same luminosity as the star that the planet and moon orbit the light it gives to the moon should be less than 0.04 times the light of the primary star.

But of course the other star could be the more massive and brighter star in the star system, and thus it might be possible for it to give to the moon a lot more than 0.04 the amount of light the first star does.

So sometimes a place on the surface of the moon might be illuminated by the near star, sometimes by the far star, sometimes by both stars, and sometimes by neither star, thus being in night. And sometimes that place might be partially illuminated at night by light reflected from the gas giant planet, and sometimes not.

The rotational period of the two stars relative to each other would not be an even multiple of either the moon's orbital period around the gas giant planet or the planet's orbit around its star. This will make the light-dark cycle on the moon more complicated.

It is not certain that the two stars would orbit around their center of gravity in the same plane that the planet orbits its star. The two orbital planes might be highly titled relative to each other. And that might make the light-dark cycle on the moon more complicated.

And if you want to make light-dark cycle on the moon even more complicated, you can make one or more of the two stars a close binary star itself, thus adding one or two more orbital periods to consider.

If you make one of the stars in your system a close binary, sometimes the two stars will be seen close together in the sky, and sometimes they could be maybe five or ten or fifteen or twenty degrees apart in the sky, meaning that one would rise or set sometime before the other. And it is possible that the brighter star would sometimes eclipse the dimmer one, or the dimmer star would sometimes eclipse the brighter one.

So if you make both stars in the system close binaries, a place on the surface of the moon will sometimes be illuminated by all four, by three, by two, by one, or by none, and the varying apparent brightness of those stars will mean that some of them will make the day much brighter and warmer than others.

So I hope my suggestions show the way for you to design a a star system where a habitable moon has a very complicated light and dark cycle, one which can be calculated and predicted but which is very complicated and hard to calculate and predict.

And be sure to check other questions and answers about habitable moons orbiting giant planets.

What are the day and night fluctuations for a moon orbiting a planet the size of Jupiter?1