Amazon Dash is a one-button solution, if he can't press buttons yet you just need to hack it so another action triggers the "button press" event.

But how can he choose different tunes? Hope to find some kind of controller that's fun for him to use.

... and of course he can press a button :)

Have several dash buttons? Like in the article where each figure plays a different tune.

Otherwise you'd have to macgyver a toy with giant buttons, build a Bluetooth device onto it that sends a different signal for each button.

They have like a 4 second latency, which would be tough for a child.