I agree with perhaps getting a bit bigger cartridge, being more useful as your son grows, and using reduced recoil loads in it. You need not be a handloader to enjoy reduced recoil loads any more as both Remington and Federal have come out with reduced recoil loads for several different calibers. In reduced recoil load, the '06 recoils about the same as a .243, a huge difference. I'd think the two best choices might be the .308 and 7mm/08, shooting reduced loads initially and then moving up to higher power loads as your son grows/ages. Just another twist on the caliber question....The 7.62x39 has about 7lbs of felt recoil, the .243 Win 9 lbs and a 30/30 around 11 pounds. Unless you handload, I don't think there is a better option than a .243, for a young hunter. If you DO handload, you might consider a 260Rem or 7mm-08, with reduced loads, as those cartridges will offer better big game performance, as your son grows and can handle more recoil.
Just my .02