Depends on the child. I've seen 12 year olds that can handle a 12 gauge and some that couldn't take the recoil of a 3" .410.
Also, as has been mentioned, the idea is to start him out with a "sure thing" as much as possible for encouragement and positive reinforcement. Unless he's a much practiced dead shot with a .22, chasing tree rats with little lead pills and not scoring can be discouraging, as well as possibly dangerous.
Also, there is a difference in a 12 gauge semi-auto that 'pushes' against you and an old polk stoke single barrel that wallops anyone who shoots it. If he has the stature to handle an available 12 gauge SA, load it as a single shot and start him out that way. Trap & skeet loads are another option.
Even with the lowly .22LR one must be very conscious of his back stop since they can fly quite a ways with the power to injure or kill.
A lot of variables to take into consideration - too many to make a definitive blanket statement about what works best. Depends on the individual's stature, experience, ammunition used, and the area to be hunted for starters.