Try a Kyosho Mini Inferno/ST.
I have one here that I have tried to give to my 17-month daughter. She likes to play with the transmitter and push the car but doesn't link the two yet
The nice thing about the Mini Inferno is that it is 4wd so pretty easy to drive, runs on AAs so is pretty slow out of the box and you don't have to take a charger with you everywhere, is a real RC car so you can get parts, but best of all the Kyosho wheel transmitter it comes with is pretty compact and ideal for smaller hands.