All the dive centers I know, here on Ambergris Caye and elsewhere, are used to having child divers and I don't think that aspect will give you any problems. I've taken children to the Blue Hole and given them shallower dives which still show them much of what there is to see. I think your focus should be on where you'd like to stay, and then which hotel. AC is quite large with many hotels and dive operators, and can be fairly impersonal. Caye Caulker, 15 miles away, is much smaller, generally rather cheaper, and has a much more "laid back" feel. Fewer dive operators, but at least one very good one. Question is whether you'd find the quieter feel more relaxing or (for your boy) possibly a bit boring.
The other main diving area is Placencia, on the mainland way to the south. I don't know it very well but feel that you'd be better on AC. Or there are some lovely private island resorts (you may see mention of Thatch Caye in these pages) which offer a great deal. Again, a 13-year old might find them a bit too quiet, but it depends on the 13-year old!
Write back with a bit more info on the sort of place and ambience that might work, an idea of the budget, and I'll recommend further.
I should add that as your visit is over Christmas you need to move fast. Some of the more popular places are undoubtedly full already.