This planet orbits a larger planet, which then orbits a binary solar system. Is there a way - based on the tilt of a planetary axis or the positioning of the planet in relation to its mother planet and stars - to go completely without sunlight for a season? The life on this planet aside, would it be possible for the planet to literally not see either of its suns for three months? I'm thinking it could be similar to Iceland during winter, but that's much more dependent on the planetary axis.
Yes, it's possible.
As the comments have mentioned, this not only exists in our solar system, it even happens right here on Earth near the north and south poles.
For the sake of completeness, I will enumerate varying reasons for this phenomenon:
- Axial tilt - causes one side of a planet to face the sun while the other side stays in darkness. The amount and length of time depends upon the amount of tilt and the orbital period of the planet. This occurs on Earth and Uranus.
- Slow rotation period - Having a planet or moon that rotates very slowly will cause varying amounts of daylight. As a general rule, the slower the rotation is, the longer both day and night lasts. (although it really depends upon the rotation period as compared to the orbital period) This occurs on Venus.
- Shadow of primary - in your case, a slow orbit around a large planet might cause a shadow on the moon.
Consider a moon that orbits a planet above its equator. As an orbit's altitude increases, its period increases. That is, it takes longer to make one full orbit. If the moon is always above the equator, then it experiences a solar eclipse once every orbit. Given the right size of planet, at the right distance from its star(s), you could get those eclipses to be arbitrarily long, if the moon is at the right altitude. During these eclipses, it will be dark on the whole planet. Remember though, the time between eclipses will always be longer than the eclipses themselves. I think this is unavoidable.
The specific calculations are beyond me, but this seems like a possibility.
The planet would have a period of rotation of 6 months. I can't think of a real-world example for a planet with this long a period of rotation. Some rotate much faster than 24 hours/revolution (the gas giants), some are tidal-locked with the Sun (Mercury, Venus), while the larger planetary satellites are likewise tidal-locked with their primary planet, leaving Earth & Mars which rotate very close to once every 24 hours for some reason. I'm unsure if this is coincidence, or the default rotation period for all planets.
My guess is that as long as your fictional planet was far enough away from its stars -- at least the distance of the Earth from the Sun -- it could have any length of rotational period you desire. Which it would need to have to maintain a stable orbit around a binary star.