When my son was 12 I was looking for a gun and caliber that would be suitable for him.
I talked to an old gunsmith and he recommended something that had never occured to me.
He said to get a caliber and gun that he could use when fully grown.
what he suggested I went with, we bought a remington model 700 in .270
He ordered a moderately priced extra youth stock and down loaded hand loads so the recoil was much lighter.
My son used the gun set up that way until he was 15, we then put on the origional factory stock and he went to factory ammo.
It worked out great, he is mow 33 and has an 8 year old son that he plans on giving it to to use as he did.
Most any caliber would be good, you could even use a 300 win mag if you wanted to and do as I did with the youth stock and down loaded ammo.
Hope this helps
Larry