well there are many ways to go om this one. i am now 42 but had my first mopar a 68 charger 383 at age 15. i payed for the car with my own cash, and had to fix it up. body work, paint, mechanical stuff, interior. i kept it in nice shape and was respectful and mindful on what it could do. i also had lots of sweat equity in it.
that being said if your kid is willing to put in the sweat equity into building their first car, with some help from dear old dad. i think its a good idea. they will appreciate it more than if you just give it to them. i wouldent go crazy with an engine. make it look fast, and tune up a slant six. if they really like the car, then later on when they show they are a good driver you can maybe build up an engine with them, another father / kid project.
another idea would be to troll the local city auctions. you can find used police or detective cars. usually crown victorias, or chevy Coprices. these cars were built for severe duty, and the cities usually take care of them well in the mechanical department as they have to always be able to run. they may not be perfect body or interior wise but they can be cheap wheels.