Budget is at least the price of a Tikka I'd imagine. Plus I agree that if he reloads that a .243 would be a good option. However I've recently been on the same road the OP is traveling with my daughter who will be ten in Jan.
What we perceive as almost non-existent recoil is a lot different than what an EIGHT year old perceives. The worst thing you can do is put them into a rifle that they can't tolerate to shoot. My daughter couldn't handle a .223 until I loaded them down to cast bullet levels and put earplugs inside of her earmuffs. The report of the rifle was bothering her far more than the recoil. Now that she has no problems with a .223, I'm moving her up to a .250 Savage so when she is old enough in a couple of years she has a rifle she can hunt Colorado since they don't allow .224 caliber rifles for big game.
That said don't make the mistake of starting them on too big of a rifle and then have your boy not want to shoot it. If kids don't enjoy shooting a rifle they won't and you'll have hell keeping them interested in it. Another thing is don't be afraid the .223 or .22-250 isn't adequate for deer, more states than not allow the use of .224 caliber centerfire rifles than not, and with bullets like the TSX and Partition you can take advantage of these light recoiling cartridges.
Another thing is since the OP mentioned "hill country" it makes me think that he is in TX. If he is hunting in TX more than likely it is going to be over a feeder from a blind with his son. He can very likely control the distance as well as the types of shots he will allow his son to take. This probability negates any real advantage the .243 might have over the .224 calibers by simply having a larger heavier bullets.