Sadly, it's not possible to overnight right in Fishers Island, NY Airport (FID-Elizabeth Field). However, you'll definitely find a comfortable place if you venture further afield.
Unfortunately, you won't find any accommodations within close reach of the runway. But if you don't mind taking a short road trip, the Hilltop Express Inn is a good option. You'll be comfortable here, but make sure you leave enough time to get to the airport.