Monokuma's name (モノクマ) is a pun on the word monochrome borrowed in Japanese (モノクロ, or sometimes モノクローム). As you can see, all but the last characters match in Japanese. Presumably that's why he's (almost) totally white and black (monochrome).
As for whether the different sides mean something, in the original game when he talks to you, which side is mostly facing you changes depending on what he's saying. When he's saying things that are especially violent, suggestive, or otherwise evil/despair-inducing his black side is primarily visible. When he's pretending to be a nice good headmaster bear, he'll be facing you with his white side, often holding a flower. A lot of the time for just ordinary talking he'll be facing strait-on. This is very obvious in the game, but somewhat less obvious in the anime as it seems to be hard to animate the switches. I do not think that the fact that his belly button is white is in any way significant.
Here are images of these three cases from the game. He makes a lot more expressions in the strait forward position, such as his angry or nervous expressions, but for brevity I've only included one:
This answer may be incomplete, since it doesn't give an in-universe reason for Monokuma's color scheme. If any reason exists, it would likely be explained in the prequel novel Danganronpa/Zero*, which I haven't read.
*As a note, if you want to avoid having the ending of Danganronpa spoiled, I would not recommend looking for information on Danganronpa/Zero, as it will spoil nearly everything.