Man, I'd just totally alienate the hell out of her. If you don't want to do the legwork and find good music, you can have fun with your bad music and it's no real skin off my nose.
That said, the best alienation possible:
Merzbow, Pulse Demon
Orthrelm, Ov
Plastikman, Closer
The Shaggs, s/t
Jerome's Dream, Completed 1997-2001
Crass, Do They Owe Us a Living?
The Pop Group, Y
a good recording of Luciano Berio's "Sinfonia"
Sunn O))), The Grimrobe Demos
Pharoah Sanders, Karma
Of course, if you ever want this girl to speak to you again (and if I don't want to be an asshole and make offtopic posts) maybe a different list would be in order. I also put reasons behind each of these, because rather than taking the list wholesale you'd be better off just using the rationale behind it.
Pavement, Crooked Rain, Crooked Rain
- exposure to the Matador catalogue, indie rock classic
The Flaming Lips, Transmissions From the Satellite Heart
- band with a large output, fairly accessible yet noisy, exposure to Restless catalogue
Slint, Spiderland
- indie rock classic, exposure to Touch & Go distro
June of '44, Anahata
- decently accessible post rock, further exposure to T&G distro
Sonic Youth, Daydream Nation
- good jumping-on point for SY, accessible yet noisy
Gang of Four, Entertainment!
- similarity to something she may have heard, if she listens to the radio; classic post-punk
Matthew Sweet, Girlfriend (or Altered Beast if you're nervous about dropping unintended hints)
- important and accessible pop
godspeed you black emperor!, F#A#[infinity] (can't find the alt code for infinity)
- gybe!'s best, exposure to Kranky/Constellation labels and the Montreal collective scene, relatively accessible and unabrasive
The New Pornographers, Mass Romantic
- accessible as hell, exposure to Mint Records and hopefully to related bands, ie. Destroyer, Zumpano, Neko Case
Built to Spill, Keep It Like a Secret
- melodic and accessible, exposure to NW scene
In short, go for accessible over best/important, because this girl might not be prepared for really difficult listening; ease her into it. Also, it's not a bad move to angle for bands that could lead her into other bands and scenes if she liked them enough to look into them. Indie labels love sending out catalogues, maybe get her a few of them to browse through.