Automating music box with electric motor
You will need an electric motor fitted with a simple gearbox: An example would be this one:


Then your job is to tinker an mechanical connection to the crank of the music box and provide 6V to the motor. A 9V battery and a linear regulator like the LM317 will do the job; your circuit will look very similar to this:



When you mount the motor into a case, you might want do decouple it from the case so the sound of the motor doesn't get amplified by the case bottom. So don't just bolt it down but use some kind of rubber feet.

Update: I thought about other sources for electric motors and came up with the idea of gutting an old cassette player: The bigger ones (read: not walkmans) used to have a flywheel which would provide a very smooth rotation of the gears (obviously to avoid variation in tape speed which would alter the pitch of the sound). Mounting such a thing an connecting could be a little bit more tricky, and I don't know how much current these motors need (especially when the heavy flywheel is standing still).