My first shotgun was a beautiful Italian single barrel .410 folder at age 15. I actually loved that gun. But when you are young, you really want to, and do love your first gun. I soon realized it's limitations, and traded it in for a beat up Baikal single barrel 12. I hated that gun. It kicked hard, and used to batter my middle finger with the trigger guard, and hitting anything with it was not fun at all. I traded that in for a nice used Zabala Hermanos (Spain) double side by side 12. It had 28" barrels, checkered solid rib with brass bead front sight, double triggers, modified and full choke, really nice wood - beavertail forend, silvered engraved receiver - nice fit and finish (was probably a 1960s manufacture), manual ejectors. I could hit almost anything with that gun even a long way out. At close ranges I was careful to allow rabbits and ducks to make a little distance before pulling the trigger. It was a pleasure to shoot. However, it weighed about 8 pounds IIRC, and over time I grew weary of lugging it around on long outings.
I traded in the Zabala for another brand new Spanish gun - I do not recall the make - a light double 20 side by side, 28" barrels. It had plain wood, blued receiver, noticeably poorer fit and finish. And what a mistake, tthings were just not the same. While it was an easier carry, it was a little sharper on the shoulder; not uncomfortable, but it did buck noticeably sharper. But more importantly hitting bunnies and birds became a challenging proposition. The weight and balance of the Zabala 12 had been conducive to a prefect swing when needed, and steadied it on quartering or rising birds etc.Combined with the denser patterns the 12 was a far better game getter.
Because of your child's age I would not recommend a 12 gauge, or a 20 gauge. I would go with a .410. One of the important things the .410 taught me was the need for some precision even with a shotgun. Used within it's pattern density for shot size limitations it works fine for rabbits etc and wingshooting. It is a good learning tool. When you feel the boy has the skills aquired, and the physical stature to wield a 12 gauge he can make the jump. I would highly recommend a side by side on the heavy side with 28" or 30" barrels, double triggers, and chokes to suit the distances game is to be taken. Improved cylinder and modified, or modified and full.