You didn't tell us how big your boy is.

Have him try out your deer rifle and see how he reacts to it. If the recoil bothers him then get the smallest round for him. If he develops a fear of recoil, then he will also develop other bad habits in shooting.

If he is a stout boy like I was at that age, then the 7mm-08 or .270 will be fine for him.

Another option is to get one of the larger calibers and then add an aftermarket recoil pad if necessary. These usually reduce the recoil a fair amount.

Personally, I would worry about the fit of the rifle first and then worry about the caliber. Go to a gun shop that has a large selection of rifles and have him bring it to his shoulder. Ask him which one feels better, comes up to a shooting position better, etc.. Once you have done that then see what calibers are available. The .243 and .270 run neck and neck as to the popularity of the three calibers mentioned. The 7mm-08 is rarer and will be more difficult to find shells for.

Good luck!
Matt