Get a decent "youth model" 20-gauge pump with an adjustable stock. I'm a big fan of the Mossberg 500 Super Bantam. If they're not yet big enough to lift, aim, and fire a 20-gauge loaded with light trap loads, they're not big enough to be shooting a shotgun, period.
Watched Andy Devine on a hunting TV show about a 100 years ago (he died in 1977) drop bird after bird with one. He looked like he was shooting with a child's toy.
I dunno. I harvested my first two whitetails and a raft of ducks with a bolt-action .410 shotgun, long before I got a Remington 870 in 20 gauge. Now I like 12 gauge, but I'm not about to underestimate the .410