We celebrated my DD's sweet 16 in NYC. She was able to bring one friend. We took the train to NYC and the Mega Bus home. Now we did stay at the Waldorf Astoria which is not budget friendly and used a car service for some of our outings.
For her birthday we:
Had her pick a restaurant that she wanted to eat at.
Bought tickets to see a NYC play.
Bought tickets to see the Nutcracker at Lincoln Center.
Did 5th. Avenue and shopping.
Spent a full day in Times Square.
I wrote letters ahead of time and we were in the audience for the Regis and Kelly show and also the Conan O'Brien show. (this was a great idea and free).
We did Rockefeller Center, the lights, the tree, and ice skating.
Visited China Town and Little Italy.
We ate at Otto (Mario Batali's restaurant), Ellen's Stardust diner, Carmines, Rosa Mexicano, Oscar's Brasserie, and tons of little deli's and hot dog/hot chocolates from food carts.
I will not lie, it was not a cheap birthday present, but she has told me time and time again that she would never change her mind, it was her favorite birthday/vacation/time with friend and Mom she has ever had. It was the best bonding we have ever had (even with her best friend and my best friend being there). She said she felt like a Princess the whole trip and could never thank me enough.
Now here's some suggestions:
See if any of your friends have hotel points or perks they can offer to you for the hotel room to defer the cost there.
Use
www.broadwaybox.com for tickets to the show. The ticket booths have a lot of restricted view ones and seriously we found a better price checking and checking through the above link.
If you are in a city that has mega-bus, book the second your dates open up. You can get $9.00 fares one way if lucky, I got mine at $21.00 per person, still very economical.
Pick out restaurants that will be more classy than she is used to, but still in the clique zone. Any of the food network restaurants will fit, and let them get dressed up and feel like little socialites. It is really fun!!
If she gets gifts from family ask for gift cards to stores like H&M, or stores you do not have in your area. This way she will be able to shop and get the great after Christmas deals.
Check out the Arts in NYC - the Museums, Lincoln Center, etc. These are great ways to show her something she may otherwise not be able to do until she is out of college.
Let her splurge and heckle for a price in Chinatown. Yes it' a knockoff, but the experience is really fun. This can be a wallet, jewelry or even perfume.
Last but not least, go to Central Park. Dress warm. They have little stands on some of the weekends during the winter even though it is cold. She got her favorite hat at one of the stands, and loves it even to this day.
Last but not least, even though it's cold if you do not have a Pinkberry in your city, GO!!
Hope this helps.
Now, IT was cold, slushy and we changed clothes a lot (her birthday is December, so you will be in the same boat).
... It is VERY COLD here in NYC in Jan but the suggestions above sound just about right . TKTS is great I suggest a great girls play is Momma Mia the music is good and the kids will like it , Mary Poppins is good as well as the Lion King I do love the old stand by of Phantom !! I am sure all these shows have TKTS tickets . 
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