The OP doesn't mention handloading vs factory ammo, and doesn't mention what the boy will be hunting, so...
First things first...
Have him try some rifles chambered in various cartridges to see what he can handle safely and comfortably. Buy some ammo and have him try your rifles, or those of friends so ha can see what he prefers first hand. He may surprise you and can handle the 30-06 just fine. If he hasn't fired it before, work him up to it with some less powerful cartridges first, or buy some lighter loads for him to try...150 gr ammo or reduced loads if your lgs has any.
If he is comfortable with the rifle and cartridge, he will enjoy shooting it more and practice more, which will build skill and confidence.
I did this for my daughter when looking to get her first rifle. We tried the 260 Rem, 7mm-08, 270 Win, 308, 338 Federal and 358 Win (BLR which she used for her first deer) in various bolt and lever action rifles, of mine and a friend's. We worked her up through the various calibers as she tried them all.
In the end, she chose the 270 Win as the rifle in a Rem 700 Titanium and cartridge as the one she preferred to shoot. And she could produce 1" groups with the 130 gr ammunition in this rifle. (While not an inexpensive rifle, I had the budget for it and it will last her a lifetime.)
She was happy with the felt recoil, as well as the weight of the rifle package itself in handling and shouldering it, and firing from the bench and other shooting positions. In the end, I think the weight of the overall rifle was also a determining factor for her (she is just 5'4").
It was available as a used rifle and scope (VX-3 3.5-10x40) at the lgs that they let us try, and I bought it at a good price after that. (She has now taken moose and sheep with this rifle).
But have him wear ear plugs and ear muffs...many inexperienced shooters "flinch" as much from the bark of the rifle as they do the actual felt recoil.
Pick an appropriate cartridge that will handle the targeted game that he is comfortable with...if only hunting coyote, black bears and deer, the 243/6mm will be just fine with proper bullets. If only up to and including Canada moose, the 6.5 in any flavour will work fine, given reasonable distance and shot parameters.
If adding elk to the picture, a 270/280/30-06 will work just fine if he is comfortable with them...and only he will be able to demonstrate and/or tell you if he is.
If not, hold off on longer shots on rutting bulls until he is older and can handle a suitable cartridge for this scenario.
Some small people shoot large magnums very well, where others cannot. Only practice will let you know where he is at.
Find a rifle that fits the lad and balances well for him, and isn't so heavy that he cannot handle it easily and safely. And if it doesn't have a good recoil pad, get one. And/or get him a PAST recoil pad he can wear on his shoulder.
For an inexpensive rifle, the Ruger American series rifles shoot pretty well. But it really depends on the OPs budget for this.
A good used rifle may also net him what he is looking for.