OK, I live about two minutes from the airport.
First, the airport is not actually in Providence, it is in Warwick.
You'll want to know this only because if you have an early am flight, you'll want to be close to the airport.
The airport hotels are mainly on Post Rd and Jefferson Blvd (runs parallel to Post Rd)
If I were going to stay somewhere, here are my top 3 choices:
Holiday Inn Express/Jefferson Blvd
www.holidayinnri.com
La Quinta Inn/ Jefferson Blvd
Best Western Airport/ Post Rd
All three get good reviews on trip advisor and all 3 are about the same price.
On Post Rd, I would check out Dave's Bar and Grill, Chelo's, or Bertucci's for dinner the night before.
Google stay, park and fly T F Green. I believe all three offer breakfast and two weeks parking.