I do, but it's usually just when I'm bored or browsing something or being mildly contemplative. It's usually some random melody from a song that I've changed a bit so I'm not getting that song stuck in my head but at the same time I have something to occupy my mind. Other times it's a miserable attempt at a beatbox. :(
I don't really come up with many melodies in my head, any more. That being said, whenever I plonk myself down in front of a piano, I normally like to just play whatever comes into my head. For the most part, it's musical nonsense, but something resembling an original melody.
In terms of what I like in a melody, I'm definitely of the opinion of simple is better. That's not to say I can't appreciate a good harmony or some counterpoint once in a while, but getting the fundamentals right is key before attempting anything too complicated.
I do compose some random melodies in my head from time to time. Sometimes, a tune pops up while I'm doing something and if it's a tune I feel has potential, I try to expand on it, add in a few more bars, chuck in some fancy rhythms and chords, all that jazz. The thing is, the tune usually won't stay in my head and I'll forget it as quickly as it first appeared. I try to keep it in mind to write down those melodies when I have the time, but I don't really get round to doing it. Oh well. Some of the melodies usually are derived from other melodies I've heard before, but they are mostly just spontaneously made up on the spot.
I love to make my own versions of my favorite songs in my head. For example today I've made "Tears don't fall" by BFMV in ballad version. Also some soundtracks from anime, games, those saddest ones usually with extended notes, and using a lot of flutes, violin and harp.