Last year I helped my BIL with the same situation for my 6yr old nephew. Ended up choosing a Rem Model 7 CDL in 260Rem. I found a youth synthetic stock for it in the classifieds on here. He can bang up the synthetic stock for a few years while the nice CDL stock is safe in a closet. When he's older and can take better care of it, the walnut stock will go back on it.
I've worked up some reduced loads for it with a 100gr Hornady SP. Figure he'll move up to something in the 120gr range in a few yrs and that should handle anything around here. If he ever wants to go hunt elk someday, I'll load up some 140gr Partitions for him.