It's funny how everyone here is just vomiting up their favorite anime to suggest to a mother of at least two teenagers(Or Older, sorry, I don't really know).
Since assumption is the only thing I can go on here, She's would probably be a bit overwhelmed at these titles that launches straight into the fantastic and culture(And sub cultural) influence heavy shows.
My list is a bit more palatable for just people who aren't extremely into anime, but is looking for broader reaching stories:
Black Jack the TV Series-
I think someone mention the anime Monster before in the suggestion, so I might as well as post the original character that inspired the series(A lot of works of Naoki Urasawa was inspired and influenced by the narrative and storytelling style of Osamu Tezuka, which is kind of redundant, like saying every thing animated now was influenced by Disney). One of the seminal anime characters created by the seminal anime creator, Black Jack is pretty much like an anime Medical Mystery, like House MD, but more family friendly(At least this version is). It's broad stories can be a bit cheesy sometimes, as most of the stories are purposely tearjerkers, but it doesn't detrack away from how interesting and entertaining the story of Doctor With The Hands Of God is. It's a pretty good series and is very very family friendly. Hell, maybe you guys can watch it as a family if you don't mind the cheese factor of it all.
Kids on the Slope (Sakamichi No Apollon)-
A coming of age story set in Japan in the 60's that focuses on the friendship of two boys and a girl through their shared interest in Jazz. I cannot recommend this anime enough to people who enjoy the more slice of life style of story telling; put together by the super star team that was behind Cowboy Bebop, it has become one of my favorite recent titles and I never hesistate to point people to it. The cast are both likable and realistic in character, filled with the typical heartache and misunderstandings that come with being hormonally charged mini-adults. It's worth checking out.
The Garden of Words-
This last one is a cheat on my part since I have yet to watch it(Goddamn poor quality streamsites... I MEAN... Uhh... GODDAMN LOCALIZATION LAG!), but the director(Who's other movies, 6CM per second and Voice of a Distant Star) has made some great movies that paints these beautiful portraits of a film and just look at that PV up there. No, seriously. Watch it. It's so pretty! Doesn't the art alone make you want to watch it? I mean it's fricken gorgeous! It's a love story about a high school boy and an older woman, so I think it might be a bit risque, but from the trailers and stuff I seen on it, I would wager that it'd be worth the watch. I honestly say give it a shot, I know I will be soon.
And that's three suggestions I made just to test the waters. If you manage to find these series, see if your Mom enjoys them. If she does, I might be able to scrounge up a little more to see if she will like them too.