Cheap sleep suggestion, NYC flying from La Gaurdia at 7:10am.

Hi,

I'm New York for 3 days next week, flying into and out of La Guardia,
the flight out to Jamaica leaves at 7:10am, so I need to be at the
airport early! Does anyone have any suggestions for cheap places to
stay (the cheaper the better, but I'm pretty flexible) Is staying
near the airport all 3 nights - I don't really want to move - going to
be crazy, I've got to meet a couple of people but otherwise I have no
real reason to be there.

How much and how long is a cab from somewhere central out to an
airport hotel - are there any cheap ones anyway?

Cheers,

Jim.