Recommend motor type to turn baby rocker, into auto-rocker
How about an alternative approach? Use a solenoid.



You only need to inject a little bit of energy into the system to get it oscillating. Wind up a big coil. Attach a steel rod to the back of the rocker, and when you energise the coil, it will pull the rod into it. Switch the current on and off, and you'll cause the rocker to bounce up and down. If you time it correctly, you'll achieve resonance. The best way to do this is to have a little sensor to detect the position of the steel rod. Energise the coil when the rod is all the way out of the coil, and de-energise it when it's all the way in.  This has the advantage that it'll be quiet.