Hm, I haven't been around a six-year-old since I was six myself, so I can't say for sure. But if I had to guess I'd say kids would have to be at least 8 to enjoy AC by themselves. It's a simple game, but it does involve a lot of item-screen manipulation, the controls can be a bit finicky, and there is a LOT of text to read. That said, I'm sure she'd have fun so long as you're there to occasionally help her select a tool or explain why the creepy raccoon keeps asking her about a "mortgage."
I have heard the game is great if multiple members of a family are playing, though. You can have as many as four people living in the same town, and you're all working together to plant trees and flowers, donate funds to get monuments built, and fill out the museum with bugs, fish, fossils and art. The best part is the animal villagers will sometimes talk to you about other players, ask you to deliver messages to other players in real life, or sometimes show you letters other players wrote to them. Animals will also speak to you using words or phrases taught to them by other players, or wear designs players have put up for display at the tailor's shop. It's pretty amusing with just me and my girlfriend in the same town, so I imagine it'd be even more fun if you could get your whole family in on it.