+1 on the geared motor.
This project is simple enough that it doesn't require an Arduino. We have a commercial rocker chair for our baby and it uses a small DC motor that runs on 4x D-cell batteries. The crank is connected to the rocker via a small spring, the control knob runs the motor at different speeds. Eventually the rocker hits its resonant frequency.
If I may offer a bit of advice. We have two kids now and found that the rocker is a bit of a waste. Both boys only used it for a very short period of time. (several weeks) before they didn't need or want to be in it anymore. My advice would be to not do this project, it will be very short lived.