Cheapest would probably be the Super 8 in Jamaica, not as close to the airport as the others listed on this page but closer to the E line subway and JFK Airtrain stations at Jamaica-Archer Ave. or Jamaica-Van Wyck subway station. The Days Inn is very close to the airport but not to subway. If they don't have a shuttle service to the subway you'd have to go to the Airtrain station at Federal Circle for the Airtrain to Jamaica for the E subway or the Airtrain station at Lefferts Blvd. for the Airtrain to Howard Beach for the A subway.
Cheapest would probably be