Lots and lots of crashpads for around $200/month. There is a hotel with transportation that some folks stay at for around $60/night. I have never stayed there so someone else may know the name of it. Sorry.
Try the Five Towns Inn on Rockaway Blvd -- off approach end of 22L. Atlas and Polar crews stay there. 1-800-300-3864. was about $70. Ok place. Lots of places to eat closeby. Make reservation. Tell em your with the airlines.
-- or-- the Sunrise Inn 516-593-1600. It's a little farther away but both have van service to JFK. Price was about the same except they used to give you every 7th stay free. Not too many places to eat closeby. Make reservation.
Best Western in Hempsted gives airline rates too, but I don't know their phone number.
Pan American is located on Queens Blvd. Has van service to both LGA and JFK (although it is closer to LGA) $69 per night comes to $81 with taxes, and I believe the 8th night is free but you have to ask about the program as they won't volunteer it.
Mall within walking, Best Buyes and Target also, BKing and McD across the street.