Hi,

Without giving too much thought I think hooking up a stepper motor (or a servo that can continuously turn 360 degrees) would be first step.

That's pretty easy to do with various Arduino libraries.

There are a few videos on youtube if you search for: Model Railway Turntable Arduino

One video looks interesting and it food for thought. A bit more complex that you might need initially. 

https://www.youtube.com/watch?v=cJudmksy8-k

A few push buttons and a servo/stepper would give you a simpler turn table controller. As I said a servo instead of a stepper might be easier to hook up. The arduino servo library wiki is here http://arduino.cc/en/reference/servo