Andre Norton.
Or Zenna Henderson’s The People series.
Don’t give her any of these old books like Fahrenheit 451. They’re classics but she’ll likely find them boring.
Lois McMaster Bujold’s Barrayar series is what you’d see next to the definition in the dictionary.
I think Startide Rising or Uplift War (both David Brin) might be good for a non sci-fi fan. Plenty of adventure, some romance, some rather cool uplifted animal stuff.
Recently read, Pandemonium scans like a decent choice too.
Just on checking around what I have at hand (majority of my collection of books are at my parents) and I’ve come across ‘The Sparrow’ by Mary Doria Russell, I recall enjoying it myself and it may be a good choice being mostly a character-driven story with the sci-fi elements being more background than foreground for the story the author wanted to tell.
I also read the sequel ‘Children of God’ recently and while its a quite different book its also a very worthwhile read.
The Sparrow!! Awesome suggestion, I totally agree. Great book, true SF themes, very thought provoking yet at the same time entirely plausible.
If she is also interested in sampling fantasy, I would highly recommend “Daughter of the Forest” by Juliet Marillier.
Beat me to it.
Spider Robinson’s Melancholy Elephants was the book that got my wife interested in science fiction.
I came in to recommend ‘The Sparrow’ and ‘Children of God’ myself.
I haven’t read Ender’s Game but a big fat ditto for Eifelheim, especially since she likes historical fiction. Aliens in a medieval village and plague – doesn’t get much better than that.
I don’t read a lot of science/speculative fiction, but recently I liked Cloud Atlas by David Mitchell and Hunter’s Run by George R. R. Martin. And The Gone-Away World by Nick Harkaway. Those last two recs have neat twists.
Ender’s Game got me into sci fi. I tried some 50s hard sci fi at too young an age and didn’t want to read any more of it. I tentatively tried Ender’s Game and I’ve loved Sci Fi ever since. It’s Card’s master work. Beautiful and thought provoking without being dry or trying too hard. The plot is more important than the science or a big philosophical question (unlike some of the classic era sci fi.)
I’d go with Elizabeth Moon.
The Heris Serrrano series is probably light enough for most people to get , and its a nice intro to space opera.
Declan
The John Carter, Warlord of Mars has a huge female following. It’s all about half-dressed men fighting with swords, and John Carter is always after the Screw of the Week. Its formula is very similar to romance novels.
The Stainless Steel Rat series is very easy to follow, and the typical sci-fi tropes aren’t overwhelming to the casual reader. The main character is a Han Solo-esque figure who also has spends an inordinate amount of time chasing tail. I stopped reading it after one particularly misogynist book, but it was carefully disguised as anti-male instead of anti-female.
Hmmm…is Joe Haldeman’s The Forever War too heavy, even if she liked Avatar?
Hell, technically, does The Andromeda Strain “count” enough as science fiction, or newer stuff like Eric Flint’s 1632? What about the classics like War of the Worlds?
I can dig out the Robert McCall book I have where Ray Bradbury where he notes that, technically, Singing in the Rain is a science fiction film, if that’ll help your case any. 
As a female SF ex-fan who’s mostly moved on to other stuff, I’d strongly concur with Ender’s Game, anything by McMaster Bujold, Left Hand of Darkness,The Disposessed and The Sparrow.
Not Heinlein. Please, for the love of all that’s holy, not Heinlein.
Other suggestions from me - Passage by Connie Willis (scientists investigating near death experiences), A Fire Upon the Deep or The Peace War by Vernor Vinge
C. J Cherryh. Either Downbelow Station or The Pride of Chanur to start, then the rest of those respective (interrelated) series.
I personally like Sherri S. Tepper, but it depends on if your friend has a feminist streak or not.
I think some Banks is actually very accessible to non-genre readers, but avoid some of the more … *outré *works like Feersum Endjin.
Second Brin, esp. Startide Rising.
I third… forth? Bujold. The first book of hers I ever read was A Civil Campaign. I wouldn’t necessarily recommend starting with that one, unless she loves Regency era romances. 
But Miles is a wonderful character.
Kage Baker would also be great choice–not really traditional science fiction, but her Company novels are lovely, character-focused stories. I was really sad to hear about her recent death.
It might be easy to get her started with some short stories–Ted Chiang is an extraordinary short story writer, and one of those “science fiction” writers who is also a genuine literary talent. His prose is gorgeous, and his imagination is extraordinary. Not all of his stories could be classified as sf (unless it just stands for “speculative fiction”), but they are all worth reading. Seriously, the man has only published 11 stories in his life and he’s won 3 Hugos and 2 Nebulas.
I have to agree–don’t start with Heinlein. Well, maybe “The Moon is a Harsh Mistress.” It was one of my first grown-up science fiction novels and I loved it. I haven’t read it in a while, but I didn’t find his female main character too offensive and it is undeniably a classic (but not the boring kind.)
I actually agree that Banks would be a good choice, but maybe nothing too terrifyingly gory or war-centered. I would say “Look to Windward” (fascinating alien cultures), “Inversion” (admittedly, the main thing I remember is how funny the ships are) and “Player of Games” (brutal but fascinating) would all be good choices. They really aren’t hard science fiction in the traditional way–and if your friend likes police novels I doubt she’ll be bothered by violence.
Personally I would put the kibosh on anything too hard (Vernor Vinge) anything too horrifically depressing (The Sparrow, et al) and anything too much in a specific niche (the Stainless Steel Rat, John Carter, Elizabeth Moon, etc.)
She might like Alan Dean Foster’s Midworld, it has similarities to Avatar. His Flinx series, starting with The Tar-Aiym Krang, the Tran-ky-ky trilogy starting with Icerigger, and the stand alone Cyber Way are also good.
Mirabile by Janet Kagan is also nice ecological/biological themed sci-fi.
How about some Philip K. Dick?
Left hand of darkness is a good one if she’s into feminist literature.
One thought is what if she’s not into “heavy” sci-fi?
Most of Michael Critchton’s work involves some sort of technology currently unavailable today but used in a modern setting. For example, Andromeda Strain is about an alien virus attacking a midwestern town, Jurassic park is about cloning dinosaur dna.