I'm going to second (technically third) the Cowboy Bebop suggestion. Yea, it is action oriented, usually at least one good action or chase scene in each episode, but it seems to appear to a lot of people who generally aren't into most anime, from my experience. Story, music, animation, all top notch, and lacks a lot of the turn offs that many anime have (particularly Shounen).
FLCL is wonderful too, but it might be a bit too weird to try to appeal to someone who isn't all that into anime. It is a great series and has outstanding style, but it just probably isn't a good place to start.
Samurai Champloo might do for the same reason Bebop can, although I always felt in general that it felt too much like Bebop, and ultimately was like Shinichiro Watanabe paying homage to himself, which made it all feel kind of weird.
Paranoia Agent was also mentioned, which has very little action (except some towards the end when the show goes so far off the deep end it's amazing). But it's an interesting, weird, and short anime.
And Death Note as well as Monster may also be worth looking into, although I have mixed feelings on Death Note, and have not actually seen Monster, but always hear praise of it. I give Wolf's Rain a nod for it's sheer beauty, I loved it's look and feel, but thought the series as a whole fell short.
