I finally managed to find an auto-rocker setup using a servo, a spring and a piece of bungee cord.
The key was finding something flexible enough to allow the right back and forth motion.
At this point I see no need to add a sound trigger, as my goal is simply to give myself a little more freedom from having to manually rock him to sleep, as opposed to something like a robotic nanny.
As much as the current setup seems to work, I'm guessing somebody with more of a background in physics could probably achieve better control of the rocking motion with something like a solenoid?