According to how much you want to put into it, there are some very economical options. Here is one I am intimately familiar with,
The Yildiz Elegante. It is available in any gauge you wish for her, barrel length as short as 61 cm (24") and weighing 5.3 lbs. It used to sell at Academy Sports for $389. Available with to without ejectors. A friend bought me one as a gift about 12 years ago, just because he knew I like .410 S X Ss. I was, and still am, amazed at the value for the $$$. All I did to it was lighten the trigger pull (on it and another friend's like it). Very easy to do, and I did it myself on both guns, successfully. The single trigger has never malfunctioned in any way and the trigger pull has stayed right at about 4 lbs. Never a hiccup with these little lightweight guns for me or my friends (we own three of them). Great thing about it is that, with it being priced so reasonably, you won't feel bad about cutting the buttstock back for her, then adding the piece back later as she grows. Comes with choke tubes now, my .410 didn't.
I wouldn't hesitate a second to buy another one for a grandkid to start out with. And I own and appreciate fine doubles as well, lest anyone think I don't know quality, or junk, when I see it. I know we're imprinted with the idea that low prices are always indicative of low quality, but these little guns slipped under the radar.