Awesome age to gettem involved in hunting and shooting. Either of your options is great. I can only tell you what I'd do and try to explain why.
I'd go with the 300 BO and here is my list of reasons.
Inside of 150 yards, the larger diameter bullet may provide more blood trail in the case of poor shot placement, which is very likely with a nervous excited youth.
Buy a few boxes of subsonic ammo. Sound accounts for just as much flinch as recoil with kids. A quiet gun is far more likely to get practiced with than a loud boomer.
Theres a lot of ruger American take off stocks layin around for almost no money. Easy to cut cut off and reattach recoil pad to make your own youth stock. This is also a great project for you and your son to do together.
The 110 grain Barnes tac x ammo is great for whitetail, Speaking from experience with my son.
If you start back reloading, hammer bullets will take that blackout to another level of killing power.
"Normally ". You can buy cheapish bull fmj blackout ammo. In a couple years your boy can set in the backyard and blast away at milk jugs and pop bottles without dollar signs flashing in your eyes at every shot!
If you decide to use the ruger and have trouble finding a stock, let me know.
Again, either option is great.
The only definitely wrong answer would be a creedmoor