Good God. Those are the exact same 3 series I was going to suggest after I read the first post. How eerie.Originally Posted by Mae
Ghost in the Shell and Paranoia Agent are akin to American live-action thrillers (one a cop show, the other a psychological whodunnit), so she'll be able to enjoy it without being "into anime". They're stylistically similar to any drama you'd find on American TV, in my opinion.
Kare Kano is a godly shoujo series (and actually my favorite anime series ever). It's got a great mix of comedy and drama, so it might be enjoyable to a teenage girl. The first episode isn't too heavy on Japanese cultural references or the viewer's knowledge of anime cliches, so it's a good gamble to start with that.
You can also never go wrong with Bebop. It requires little knowledge of Japanese culture, and is pretty realistic. That is to say, it's not too "out there" for a Sci-Fi series. It's perfect for anyone just starting to watch anime. Like others have suggested, any anime with emphasis on story instead of weird gimmicks would be great choice. I've only seen one episode of Monster, and I agree it seems OK.
Stay away from series with obscure Japanese cultural references like Azumanga Daioh (however funny it might be, I had to read a fuckton of info before I understood even half of the jokes). Those will bore her out of her mind. Stay away from anything an over-the-top Otaku would watch (laden with anime in-jokes or anything that would require an open mind to watch - Excel Saga, any Mecha series, Haruhi Suzumiya, any Magical Girl series, or any "Harem" series like Sister Princess, Love Hina, Tenchi, Maho Sensei Negima, etc) Also stay away from stuff that might freak her the fuck out, like Perfect Blue or Evangelion.
Good luck. I'm actually interested in what you end up picking.