I have both a .243 and a 6mm Rem. But the .243 has killed a lot of deer.
I don't know how big your son is, but when it concerns recoil, I would start him on a ,243. I started all 5 of my boys on the .243 and can't recall a single deer walking away with a normal and well placed shot. 90 gr. to 105 gr. bullet will take care of business quite well.
I have a longtime friend who is a guide, that is blind in one eye, thus he doesn't like to shoot higher recoiling rifles. He has killed virtually everything, including deer, antelope, elk, black bear and big horn sheep with his.
One of my DIL's who just started hunting 3 years ago has successfully killed her deer every year with her little .243 shooting 90 gr. Hot Cores.
I'm not very familiar with the 7-08, but I would imagine it probably packs a bit more recoil than the .243.
GS