My vote is for a 7mm-08, but in fact most calibers can be tamed to low levels of recoil with the right ammunition. A 300 Win Mag with mild loads can have recoil equivalent to a .243. Starting out with a 7mm or larger caliber will give you more options should the gun be used on larger game such as elk.
Of equal or greater importance is finding a gun that properly fits a small shooter.
Lastly, don’t overlook the importance of proper ear protection. I’d encourage the use of foam plugs and muffs. I believe more flinches are developed in new shooters from anticipation of a loud bang than recoil.
My son started with a 7mm-08 and killed quite a few critters with it. When he’s not looking I like to sneak out with it.
