I'll throw in a couple of things I have found while dealing with my three grandsons, these two quotes are big ones,
I'd let him hold and shoulder as many different models as the stores around you have. Fit is just as important as caliber. I'm a bit taller and heavier than average, and even to me a Remington Model 7 or a Winchester M70 Compact just shoulders and feels just right.
Whatever you get him...buy him enough ammo and make enough time to get him on the trigger until he's really comfortable with it.
No matter what caliber you decide on look for something with a shorter lenght of pull.
Also look into the long eye relief scopes, one offered by Weaver is great and I have two of them mounted up for the kiddo's, just don't remember the numbers off hand and they are not here to look at. Remember a longish stock shouldered by a youth with short arms and not much neck to stretch into the scope isn't going to fit well at all. If it doesn't fit even a lighter recoiling round can induce some not so fun impacts to collar bones and such.
Ammunition is another consideration you should look into before you purchase. Find out what is locally available on a sudo regular basis. If you handload your almost set if you can find you some H-4895, and some mid weight bullets for caliber that will suffice for hunting deer. Say for instance with a 243 the 85 and 90gr offerings might give a touch less recoil than the 100's do, In the 6.5 and 7mm calibers, it would be the 120 - 130gr offerings, and in say .308 those from 125 to 135 would be the ticket. Any of these can be handloaded down to levels which make shooting them pleasant as well as accurate and deadly on game.
The biggest factor as mentioned above is practice. I cannot emphasize this enough. If they are not steady and sure when they get out to hunt the results aren't going to be nearly as good as if they were rock solid in their shot. If they haven't practiced enough they will jerk triggers, squint when aiming just at the wrong times, and sometimes simply close their eyes before pulling the trigger in anticipation of the rifle going off. Even if they know it isn't going to hurt them, they still do so.
I have three grandsons ranging in age from the oldest which just turned 12 to the youngest which is 6. The oldest used my Ruger Compact chambered in .308 to drop his first hog two weeks before his 4th birthday. He held the rifle, he aimed it, and he made the shot all with no more aid from me than setting him up on my knee so he was high enough up to rest the rifle. (This is also something you have to consider as you cannot loo through the scope to help them they must be able to do it on their own and practice to do so.)
My middle one is how can I put this, more talk than he can back up. He really does want to hunt and shoot, but he doesn't like the recoil, or the amount of noise when the trigger breaks. I have had him in both ear plugs and ear muffs, to counter the noise, and he only wants to shoot one round or two at most from his .243 with as light a loads as I can get away with.
The youngest, heck he'll jump in behind my .308 with full tilt loads once, but then he is done. With the lighter .243 loads he is good for about half a dozen but he simply isn't big enough yet to fit even the smaller Compact rifles. I guess we don't call him "Tank" for no reason.
Anyway, you need to keep the fit of the rifle, and the eye relief of any optics you purchase as a top priority as well. IF it fits, and has enough eye relief, it will make shooting even something with a bit of a kick MUCH easier on them, and they will be much more accurate, and enjoy it more as a result.