@MNb
I hadn't thought of the Cochrane; that's definitely a good line for teaching gambit play and both long-term and short-term compensation. 4.Nf3 Ne4 5.Nc3 is quite simple but definitely not as exciting
@Vass
The Four Knights doesn't have to be closed; White can play 4.d4, the Scotch Four Knights, with an open position almost by force, as I mentioned in a previous post. I think that 'too much theory' is a poor argument against teaching a young child the Scotch Game - most of the time the most played moves are the best. The Scotch Game leads to open positions but the ideas of the Scotch Gambit (for example) are a fair bit simpler. I disagree with your reason for rejecting 4.Ng5 - at a low level White only needs to know a small number of lines to be successful, but a better reason is that it requires White to defend quite accurately in most lines - while being able to defend is just as important as being able to attack, this is a much more advanced skill, and at all levels but especially when one is starting out it is much easier to attack than defend.
Actually 2.Nf3 Nc6 3.Bc4 could be a good alternative to the Scotch Gambit for a beginner, intending to meet 3...Nf6 with 4.d4 (which in fact transposes after 4...exd4), and 3...Bc5 with 4.b4 as the Evans is a good opening for teaching the basics of gambit play, rapid development and playing with the initiative. But the child only needs one good line to 1...e5 and if he's happy with the Scotch Gambit I'd definitely stick with that for now.
@Smyslov_Fan
For most children I would be reluctant to have them play the King's Gambit because it could train the bad habit of them playing dubious, 'trappy' lines. However teaching them the basic ideas can only be good for their tactical vision and feel for gambit play. There is one small issue though - how do you explain to an inexperienced player that after 2...exf4, 3.Be2 is the only move that draws, and that after 3.Nf3 only 3...d6 4.Bc4 g5 5.d4 h6 wins and the main line 3...g5 4.h4! is a draw? Of course it's possible to oversimplify things and say that 3.Nf3 and 3.Bc4 are the main moves, but I don't like lying to people I teach.
As an aside, I think Vass overestimates the theoretical knowledge of his child's opponents. It's rare that you find someone below 2000 with a very solid knowledge of theory, and those who possess it tend to struggle OTB once the game leaves their book. Around 1400 level you'll see a lot of people meet 1.e4 e5 2.Nf3 Nc6 3.d4 ed4 4.Bc4 with 4...h6 or 4...Bb4, for instance.