In that case, I would go with a youth 20 ga. I started our son on a 410. but we transitioned over to a 20ga this year. He is a runt. He is only about 4.5' tall and weighs about 65 pounds. He can handle a youth sized 20 ga just fine.
There is a big difference between a full sized 20 and a youth sized. I would take here to a gun store and let her try on some of the youth models. As someone already mentioned, the Remington 870 youth would be a good choice, as well as they Mossberg 500 youths. There are other good ones as well, but may take you over your budget. A youth 410 and 20ga will weight about the same in the same model, so you really don't lose much weight by going to the 410.
I'm a big proponent of starting small kids out on 410s(many people are not). But in this case, I'd really look at the 20s. Many people will tell you that a 20 has as much recoil as a 12. I don't buy into that. Especially when the gun fits you well.