first question, where does your mom get her ideas about anime? this is not a snarky question, is she watching Cartoon Network and maybe catching some Naruto or Inuyasha? it is true that most of the anime that makes it to TV is just a bit on the violent side.
second question, where do you plan to get your anime? buying in stores can be expensive, and most DVD rental places don't carry the good stuff... maybe Netflix?
third and final question, what does your mom consider violent? could you watch a 'cop' TV show where there's the occasional gunfight with the bad guy? or maybe a series that takes place during wartime... where there is an occasional battle, but the main characters aren't that involved in the fighting. is your Mom just against the gratuitous violence like Naruto or Inuyasha where every episode is some sort of battle with knives and guns and blood?
otherwise, there are plenty of wholesome decent anime out there, some suitable for 10-13 year olds. (but a bit more anime suitable for say 14-15 year olds

).
maybe have your Mom check out the reviews on Amazon.com for some totally non-violent, not very suggestive at all anime...
"Strawberry Marshmallow" -- in the American version, the older sister is College aged, she smokes and drinks. that is the most 'bad' thing about the entire series. the main characters are around 10-12 years old, little girls doing everyday life stuff with lot's of *funny* thrown in. if my niece wasn't like 4 years old i would totally buy this for her. perfect comedy series for young girls.
(in the Japanese version the older sister is only 16 and a High School student who smokes and drinks...)
"Azumanga Daioh" -- the High School version of Strawberry Marshmallow... 5 or 6 girls and their teachers going about everyday life in a very funny way. mild violence (hitting people over the head with a book or snowball fight type violence). the series follows them through High School life with a bunch of Japanese Culture events mixed in.
you can maybe find some YouTube clips of these anime to show your Mom. warning... most of the videos on YouTube are "Anime Music Videos" and some might seem more violent or suggestive than the actual anime because people have picked the violent or suggestive parts from the entire series and put them all into one single video... but if you look hard enough you will find a few actual episodes... show them to your Mom.
YouTube for 'Ichigo Mashimaro' (the Japanese name for Strawberry Marshmallow).
YouTube for Azumanga Daioh.
YouTube for 'Aria the Natural'. i wish this were licensed and available in the U.S., Aria is the most totally non-violent, non-suggestive, totally relaxing "girlie" anime in existance. i hope it's available when my niece is old enough to enjoy it.
only slightly more adult, (you might like them, but your little sister probably wouldn't), and slightly more violent (maybe there is a war going on, or the heroines are being chased by the bad-guys and have to fight occasionally, but not really blood and guts type of violent)... but still really good 'girl-power' or 'teenager-power' types of anime..
"R.O.D. the TV" -- Read or Die.. three girls (ok, two are women but the third is a young teenager), have the magical ability to use plain old paper as weapons or as a defense. they are on a mission to solve a mystery (just like Nancy Drew). lot's of heartbreaking bring tears to you eyes moments. girl-power to the max.
"Last Exile" -- a teenage boy and girl who fly messages back and forth like the Pony Express are caught up in a war while they attempt to protect a little girl who holds the key to the planets future. fantastic animation, fantastic music, fantastic story....
all of these (except for 'Aria the Natural') are available from Netflix. hence the second question... you won't find them at the corner DVD store... you can buy them from Amazon or maybe BestBuy or whatever other store carries Anime DVDs, or you can rent them from Netflix.
find a couple of the YouTube versions and let your Mom watch. have her read some of the reviews on Amazon. there is a ton of anime out there that is not the normal violent, suggestive stuff that makes its way to televison.
this is just the stuff that hits my 10~13 year old radar, for maybe 16 years old or so there is more stuff that might or might not be interesting depending on your tastes...
"Planetes" -- life in space cleaning up junk. some bad language here and there and maybe a fight or two...
not yet available in the U.S.,
"Mushishi" -- bizarre, sorta like X-Files. no violence, no suggestiveness, but freaky and weird and sometimes people die. main character guy studies a bizarre form of life that is between say rocks and insects... sometimes people go blind, sometimes they go deaf, sometimes they find themselves lost in the woods forever.... beautiful animation, nice soundtrack, very spooky... will leave you with a chill up your spine after watching... but no real violence....
anyways, there are plenty of anime that aren't based on violence and panty-shots... but some of them are still a bit more adult themed... when you reach 16 or so, there is a lot of anime that is equal to a U.S. TV mini-series... 13 or 26 half hour episodes that tell a full story that is not full of gratuitous violence or suggestive images...