If you're able to produce an "acoustic" instrument that makes a pleasant noise, one of the clip on pickups generally used with electronic tuners (about $9 US?) plugged into an amp makes a usable pickup to let a kid simulate an electric guitar. Sound quality may be quite a lot better than you'd expect, and you can always add a "real pickup" later when you know the kid is going to stick with it - and after you've found out whether the instrument you've put together merits the cost.

A Google for "guitar children" suggests that most teachers give kids around 6 year old and up instruments significantly larger than proposed. While I wouldn't discourage anyone from DIY, looking at what "most people do" can often be a help before launching a building project.

John