As odd as this sounds: DO NOT START BY TRYING TO SHOWCASE GOOD ANIME, especially not the lighthearted kind.
I'm dead serious, though when I say 'good' I mean more 'indicative of the genre.' I've talked with friends and others about this and we've learned that starting by using the more memorable stuff can very quickly turn someone off of the show, and anime in general; usually causing them to discount anime as a whole.
Why?
Culture shock. While this seems self-evident, Japan has a different culture, which causes a difference in humour, metre, pace, tone and morality. The big problem comes from these things not being obviously different. I have a couple of examples of minor differences with massive end results resulting from this, but it doesn't make for polite conversation.
Similiarly, many lighthearted animes are bad places to start because of minor things that detract. I've gotten odd looks all throughout showing my family Azumanga Daioh despite my thinking it funny. The same goes for most 'set in the real world' animes.
Point being, if you want to introduce someone to anime remember that for many of the better or more high-concept ones you have to GET anime to LIKE anime. Which, of course doesn't mean that not getting anime is the only reason not to like it. I GET how to drink tequila, but I still don't like it.
My best suggestion is to think of the anime(s) that got you into the genre and dig them up. If you don't think those are good ones for the situation, I recommend a fish out of water or fantastic (literal sense) story. In those cases the culture shock on screen and required explanations help the viewer learn about the world they see, and learn how anime 'works' by osmosis.
Inuyasha, as a recent dubbed example, actually kind of does this really well by being a fish out of water story for Kagome and carting the other fish back with her when she goes home.
More specifically to this: I do not recommend FMA. Never seen brotherhood, but I have read the whole manga. There are a lot of little things that someone unfamiliar with the tropes might find unappealing or jarring. In the original FMA, Mustang and Winry. In the manga, her and Scar. Granted, they're deep in, but the way those interactions played out could (and has been) enough to remove people from the series.
I find for this I have the following suggestions:
Gundams are good (I like wing myself) as they are based on wandering the world and space.
Ranma 1/2, because the curses, rivals and parents are overblown by any standard and are the focus.
Nadesico: Spacefaring, culture shock as a plot point, and Ruri.
Slayers: Fantastic setting, though it did take me years to get a few jokes.
Record of Lodoss War: Also fantastic setting with much explained. For myself, I say the second series first and still prefer it. Take from that what you will.
Final point, and forgive the caps, GET A DUB. There are those who refuse to watch dubbed anime, and I don't know if this applies to you and yours, but don't start people that way. Again with the culture shock, but if you're trying to introduce someone to something and like it, it's your job to minimize that shock. Plus, the above are all fair to decent shows with good English VAs.