I went with a 22 ARC in Ruger American. I've been very happy, my son is 9, and shot his first doe and first buck last year. The LOP is 12" and fits well. I also added some of the factory weights and suppressor and the recoil is nothing. He loves shooting it, and so do his friends. Two friends have a 243 and 6.5CM. You can visibly see them tense and flinch when they shoot their own rifles. In s matter of minutes, they were all hitting a 6" steel at 180 yards with my son's rifle.
6 ARC would be great as well, maybe better. I just wanted to try the 22 and its has more velocity. I'm not a fudd, but I still think a little more velocity has its merits at hunting ranges for less drop. I just ordered some 80gr ELDx for this reason, we've been using 88s.