Hi Jenny,
I've just seen this thread. I've stayed at the Mayfair Hotel before and it is nice but I believe quite booked for the New Year already. You may wish to find a few hotels that would be suitable in price and call them directly (or email them) to be put on a wait list in case there are cancellations. I also suggest placing an Ad in craigslist.com New York - if you place an Ad asking to rent a room for the New Year next year, you might be surprised by the response. I've been quite lucky placing ads on craigslist and getting very fair deals. You will of course want to check on any deal that you make that they are a reputable agent or home owner that has rented to others.
http://newyork.craigslist.org/vac/You can post an ad or view ads, but I've found it best to just make a post describing in detail what you are looking for (without giving any personal information). You will get some good replies and likely many that are not even close to being on target, and then the most lovely part is that you will likely get a picture of a naked man or woman (or parts thereof)... just to prepare you that some of the respondents love to display themselves.

If you can tolerate a bit of that sort of nonsense then you will likely find some excellent deals.
If you want to visit a more specific and professional site, go to vrbo.com This stands for Vacation Rentals By Owner. It's an excellent site that I've used many times and found the owners renting from it to be very responsible and credible. You can find affordable deals by renting people's condominiums.
And if you end up having a spare room in a condo, please remember to contact me. I'm always up for going to New York for NYE... I'd even chip in on the condo.

If you can't find affordable flights during that time, you may wish to fly into a different city and find a flight on ATA airlines to New York from that city. ATA has flights from Chicago to New York for under $100 sometimes... so you find the best deal and then fly within the country that way.