I'm having the same problem as Stephan. I can't think of a single series beyond Ghilbi stuff. I guess MLP:FiM comes to mind but that's kinda like stealing a ball from another court.
I guess it's hard to relate to a little girl when you're an eighteen year old male... Most of the anime I watch are cute moe blobs of pointlessness, or something I'd rather not let a little girl watch, like Baccano or Lain or School Days.
I guess Shugo Chara!, Card Captor Sakura, Sailor Moon, all your cliche' little girl animes but I don't really know what they can take away from it. My little sister watched Clannad (The TV series) when she was eleven and it didn't fly over her head, so maybe there's something to take away there?
I actaully think a ten year old can watch Sora no Woto without much trouble and still take something away, though some may argue that they'll only take some sort of anti-war sentiment away from it which is really dependent on your personal views whether this is beneficial.
There is Pokemon. I mean it's cliche' and everything but it teaches teamwork, kindness, and all that jazz. All the while it still has cute little creatures to keep the child's attention.
I wonder if Macross or Macross Frontier would be a suitable choice. It's obviously something a boy would watch more than a girl, but certain elements like the romance and singing and such would be able to keep a girl entertained. I don't know, this one is a bit iffy.
Here's a question Brent: would you show a girl that age Boogipop? Or would you wait until they're more around the teen age group?
Though in the end I wouldn't really be able to tell unless I knew her better, like does she like pretty things, is she a tomboy, how mature is she, ect. ect. So if we're talking stereotypes here 'll just go with the magical girl anime (Madoka excluded).
And I would show her Ghibli and MLP:FiM weather it's against the rules or not >.>