Fender Squire do their junior models – packages like this, I’m sure Encore and Epiphone probably do similar too. Ibanez do 3/4 guitars too for little ‘uns.
The ‘how much’ question is a bit tricky. Go too cheap and you end up with something where the truss rod might need continual adjustment to make it even remotely playable, and it continually goes out of tune (well, that was my experience of cheaper guitars :P), but obviously going for something higher budget runs the risk of it being left unused if he doesn’t take to it. Personally I’d spend a bit more, then be a pushy mother, beating the child until they were the next Joe Satriani.
Considered starting him off on an acoustic?