Is this over the holidays or is this in February? Around the holidays, it is usually a lot more expensive with hotels and such. One hotel I recommend checking out is the Chelsea Savoy Hotel, it is on 23rd and 7th in Chelsea, and their room rates are pretty reasonable. They call it a European style hotel, it means the rooms aren’t that big or fancy, but it is definitely well run, I stayed there quite a bit, and looking at their website the rooms aren’t that bad, especially for a single.
There are a lot of things to do that are well known, I won’t mention those (the only one I will is the High Line, it is nice to walk even the winter).
The East Village is nice walking through, it still retains a bit of its young, hip vibe, even though like so much of nYC it has become very pricey. Walking the lower east side is interesting,very young and hip, but you can see the tenements, now luxury rentals and condominiums, that some of their ancestors probably lived in when they were miserable cold water flats and the lowest east side had the densest population in the world at one point.
A lot depends also on what you are looking for, I think that people can give you a lot of ideas if you can say what kinds of thing you like. You can rent a bike easily (they have this citibike program where the bikes are right there, you pay for it with a credit card, then peddle off).
Walking across the brooklyn bridge, if it is not so cold, is a lot of fun. There also is a transit museum in Brooklyn Heights that if you like that kind of thing at all, is fun.
Want to do something different, that takes a little bit of travelling? Take the subway out to Coney Island and try the NY Aquarium, it is a neat little place, and right down the street is Nathan’s Hot Dogs, the original, and there is nothing like a hot dog there, if you aren’t a vegetarian:).
There also is the cloisters in Northern Manhattan, it is dedicated to medieval art and culture and is a castle that was moved. They often have music there as well, that can be fun.
Again, I think that people will be better able to guide you if you give them an idea of your kinds of ‘things’, I wouldn’t want to steer you to a chamber music festival at Juilliard if you are into world music or something else;)