[JoshTriplett]

An additional possible variation: The probability of a square could determine not just whether you can successfully move from it, but also whether you can successfully capture a piece on it. So, for instance, a queen on a 40% square is much less powerful offensively, but much more powerful defensively, because 60% of the time you can't capture it. You could "hide" powerful pieces on low-probability squares, and you'll have a hard time getting them back out but your opponent will have a hard time capturing them and might not try. (You could either have capture failure leave the piece that attempted capture on its original square, or have the piece that attempted capture get captured.)

Sticking your king on a 20% square would be a great way to probabilistically buy yourself time to bring other pieces to its defense, at the cost that the king can't easily try to escape.

That has a nice symmetry to it, where rather than making 95% always "good" and 40% always "bad", they're both valuable for different reasons.