This is an interesting problem, to say the least. You're going to have to get potentiometers to ensure you are rotating 90 (or 180) degrees, instead of just a timer. You'll get more consistent results that way.
As for motors....I'm positive you could take apart an RC car and use the motors in them. I don't know the precise force required to turn a cube, but it isn't a lot, and a motor found in an electric toy should suffice. \
Best of luck!
If you really want to challenge yourself, you could make a robot that solves the cube
