First, I'll say that I celebrated my 15th birthday with dinner at Cinderella's Castle. It's a lot more expensive now, but, as far as experiences go, it's really very cool. (That assumes you'll be in the Magic Kingdom, or able to hop there from another park, and that she'd like that sort of thing.) There's just something about the castle.
If she likes the idea of foreign/adventure travel, you might try Boma at the Animal Kingdom Lodge. You wouldn't need park tickets for that, and the lodge itself is fun to visit. Yoiu could do breakfast if you prefer. If you are at Epcot, pick from the countries there if there's one that interests her (i.e. Morocco, Japan, Norway). Each country's pavillion has a restaurant. You'd need to use a park ticket for that, of course.
If she's a sports nut, you can visit ESPN Zone at the Boardwalk resort for dinner, and then wander around the boardwalk afterwards.
For a kid with elegant tastes, maybe tea at the Grand Floridian resort?
Like one of the other posters mentioned, if you want a party atmosphere, try Chef Mickey's at the Contemporary for dinner with the characters. (They also do breakfast.) Or, for a more grown up spin on the party idea, try Raglan Road at Downtown Disney. No park tickets required, the band is crazy good, the food is nice, and it will definitely have a more grown up feel. And, in addition to being at Disney, it's an Irish pub, so it's definitely geared for family members of any age.
I also like the earlier idea of ordering a cake for her. (You can do this at almost any restuarant, though perhaps not at breakfast.) The small cakes are not really all that expensive. It's also a lot of fun. Some restaurants will add things like Mickey confetti on the table if you tell them it's her birthday.
What a fun decision! Enjoy your trip!