I have already said a ruger 10/22, but one thing I need to add is a warning about weight. They tend to be heavy, especially when you add things like a new stock and scope. My son and I shot a little this weekend and he complained, after a while ,that it was getting heavy. We also shot a henry lever action as was mentioned earlier, it's lighter, but doesn't have the adjustable stock available.
I'd go with the ruger 10/22 for a semi auto, the CZ 452 American for a bolt action or the Henry for a lever action. The Rossi combos are nice too for single shot with the option of changing to a shotgun for added enjoyment. I have all these and enjoy them all. The ruger probably has the most available aftermarket accessories. You can add a scope, target barrel, trigger and stock for about another $350-$400 and have an incredibly accurate and inexpensive to shoot .22. Your daughter and you can do the improvements as a fun project together. Then enjoy the time shooting it.
i definitely agree with the 10/22, that is what i started shooting with when i was 10, my dad gave me his old one to shoot. they will stand up to almost anything you through at them and will not fail. they are one of the most accurate 22 you can get without spending 500+ and it holds its value very very well