If you are lucky and get the Holiday Inn Ariel on Priceline (or one of several other hotels in the same area), you can take the public bus for free to/from the airport. Some other hotels require that you either use the Hoppa bus (at 3 pounds per person per trip) or take a taxi, and public transit is very hard to arrange.
Depending how early your flight is, it might be worthwhile to stay in central London on the Piccadilly Line - for example at the Globetrotters hostel or the Piccadilly Hostel.