Panasonic and Hinzmann both make electric assist kits where the throttle is engaged by the pedals. Hinzmann uses a motion sensor on the cranks and Panasonic's system is pressure based, the harder one pedals the stronger the motor output.
The Hinzmann system uses a hub motor only, Panasonic makes both a hub motor, and a motor that is inline to the chain. I believe the inline system to be the better of the two. A small motor can reap the benefits of gears, giving it more effective torque, than a hub motor.
I have a Giant Lite/Twist which uses Panasonic's inline motor, my father has an Estelle that uses Hinzmann's hub motor.
Google "electric assist bicycle" and several companies sell retrofit kits.
Both companies sell their kits in whole or part.
For a 3 year old the above will be heavy, although the pedal sensors would be of use.