You have lots of options. Since LaGuardia is in Queens, you ll have to take a subway (or two) to get to Times Square. But, that is the beauty of it. You can get around by subway all over NYC. I d stay where you can afford it (even across the George Washington Bridge in Ft Lee, NJ). Good luck and watch out for the pickpockets.
The problem is most hotels in Midtown are booked for New Years. Therefore, you will have to settle with booking a hotel room in Queens or near LGA. You will be still be able to travel to Times Square, by taking public transit. Depending on your budget, may I recommend the link below.http://www.hotels-rates.com/Queens/NY/us��Good luck
The link below can help u. Good luck!note: to change the time and date go to show options
Pick a Howard Johnson.http://travel.yahoo.com/p-hotel-351176-t��Their rates are reasonable around New Years and a few of their locations have airport shuttle/bus that will pick you up for the airport.
Waldorf Histora. 3