Moe crap like K-On! and Lucky Star is out, 'cause that's "Japan's high school girl thing", and that's still creepy. In fact, I'd recommend against any high school anime. In the 90s, the popular stigmatization of anime was gory pornographic cyberpunk, but we've since shifted over to figurine upskirts and rampant pedophelia. You'll want to distance yourself from that.
Witch Hunter Robin and Ergo Proxy are good bets, since they're so undersexed they guard the Ottoman harem. They've both got a realistic, Western-influenced style; Robin is set on contemporary Earth and Ergo Proxy is SF. They're plot-driven and kind of drab, in contrast with the colorful, cute character-driven majority of today's shows. Not really for everyone, but decent enough. Wolf's Rain has a similar style and is better than either, but you might have to answer uncomfortable questions about why you're watching all these handsome young men.
FLCL is actually not a bad choice either. A coming of age story amidst suburban ennui and hyperactive aliens and psychotic robots. It's ridiculously sexual, but via innuendo rather than skin. It's coy and clever about how secretly raunchy it is and it's absolutely unique, and it's definitely worth your time either way. Since we're mostly talking about a walk-in situation, the amount of energy on screen should give your mom an epilleptic fit, so no danger there.
Cowboy Bebop. Integrates phenomenal music with sci fi noir action. Absolutely top tier anime, and highly accessible to Western audiences - I used this to get my English dad into anime. True, Faye is in it, and she's certainly very sexy, provided that she is not moving or talking or doing anything. http://youtu.be/b_K_fu-mPNU
Samurai Champloo is hip hop stuffed into anachronistic medieval Japan. And that's awesome. It occasionally uses sex for comedy, but the female lead is highly unsexy to the naked eye and is wedged between a pair of badass ronin. A great watch, with tons of action and no unfortunate implications. It also makes a good gateway drug for Western audiences, so, as with Bebop, consider trying to get your mom to actually watch it.
http://youtu.be/z-qhxMrxDz4
Here's episode one. Requires a Youtube account to watch, since it's TV-MA.
Monster. Horror/mystery by horror/mystery prophet Urasawa Naoki. A tense crime drama without moe or sex. Very mature themes, and ideal for presenting this side of anime. Also, it's completely great and will chill your very soul.
http://www.watchcartoononline.com/monster-episode-18-english-dubbed
Here's episode 18. To my memory, it doesn't need much context to operate, and it has a lot of what makes Monster good in it. Note that the security of online viewers may be dubious.
Ghost in the Shell: Stand Alone Complex. Okay, this is really mature and is probably the smartest anime I've ever watched, full of politics and transhuman philosophy, but it does have the Major in it, and she does wear whatever she wants. So it's really good, and it presents a good side of anime, but it might kind of communicate "even when they're really smart, Japan is still weird about women".
Anything by Ghibli. I like Laputa and Kiki, personally, and Totoro was my favorite movie as a kid. Combines childlike innocence with a maturity that can be taken seriously by adults. Good watching for all ages, but some can be a bit surreal for first-time viewers.
Nodame Cantabile. Shoujo romantic comedy at a musical college. Also pretty long, so I haven't actually watched all of it. At anime club, we watched this and Demonbane. We were laughing AT Demonbane, but we were laughing WITH Nodame Cantabile.
Kino no Tabi. A young traveller explores a surreal world and visits many unusual countries. Explicitly philosophical, like Ghost in the Shell or Serial Experiments Lain. Not as good as SAC by a long shot, but doesn't require explaining the Major's wardrobe.
Finally, Cromartie High School.
http://youtu.be/ceMMFjsLHlg
This is the opposite of sex.