Being a Batman Fan, I'm quite happy to see a Batman Film done right. Arkham Asylum is one of the best games I've played, and until the sequel comes, I doubt anything will top it. Batman is, well, Batman. Your kid probably loves him before this game came along.
The bad points for your kids is the fact that this game is extensiveness Violent. After what is, in my opinion, the most frighting version of the Joker takes over Arkham, the halls are filled with the dead bodies of Arkham security. And Scarecrow appears in this game, and boy does he live up to his name. Honestly, I'm sometimes surprised this game didn't get his with an M rating.
All in all, Batman: Arkham Asylum is a Dark thriller that's true to it's sources and loads of fun from beginning to end, and then some!