I love this article:
http://www.skincancer.org/prevention/tanning/appearance-t...
If your girl is pressuring you, tell her simply "No, not until you are 18 and old enough to make an educated decision." Period.
THEN, find a 'win' for her. If she's looking to spruce up her appearance, consider other things which might be less carcinogenic. Some new earrings? Shopping for a cool outfit at a vintage store? This might be the time to make an exception and go to a makeup counter and let them do her up-- she could pick out a couple shades of eyeshadow or liner, blush, a good powder for her face.
My uncle nearly died of melanoma, so I do understand your concern. I grew up in the age before sunblock under the Honolulu sun and have lots of moles and freckles. I'd acknowledge that your daughter has friends who are allowed to do this, and that you understand that she feels she's missing out. The 'just once' argument is likely to be fuel for further arguments, IMO. ("It was fine last time, nothing bad happened to me... why can't I do it again.") This is why I suggest the article above and then finding some way to soothe the urge to look just like her friends.
You may also need to think about some very real consequences for her if she decides to sneak off to the salon once she's 16, too.
ETA: I, too, second the idea of a spray tan. That's another, safer, option.