The weight of a .410 and a 20 ga is typically the same across the same models, so don’t let that determine your choice. The youth or bantam models usually have the same length of pull between .410 and 20 ga, so that shouldn’t determine your choice either. The fit of the gun and the weight of the gun will contribute to the felt recoil, that will determine how well a person “handles” the gun. You’ll need to have your young shooter actually hold the gun before you buy it, to be sure it fits, or can be made to fit properly. The 20 ga has far better ammo offerings and availability than the .410, and usually better pricing, so I recommend you do weigh that in your decision.
With any hunting situation, you have to use the right ammunition. For new shooters especially. If they become recoil-shy, they will tend to shoot poorly and lose interest. But if the loads are not adequate, the shooter may lose confidence and be disappointed.
I strongly recommend the Mossberg SA-20 Bantam. Best of everything – except availability.