there is a youth hostel on Amsterdam and 103rd. it's spartan...but clean...and cheap for new york. I think it is around $30 per night, although you may have to share a room with a lot of people. There are a couple other hostels if you're willing to go that route...but book early--they fill up. Another option is to try using hotwire to book a hotel. You can name your own price and if there is something that matches, you will get it. However, hotels in New York are very expensive, so you're not likely to find anything that will match for a tight budget.
I didn't know you could bid on prices on Hotwire; Priceline does, and it's worked for me. But the hotels they list are NOT in the $30 range. You might also look into Servas. www.joomla.servas.com