There is a Crowne Plaza in Eagan that offers a room rate that comes with shuttle service and parking. Stay the night before you leave, easy shuttle service to/from MSP airport.
There’s also a public parking facility in between MOA and airport. If you don’t want to incur the hotel expense or want to stay somewhere close you could just drive to this facility and hop the train to the airport. This is a covered ramp facility.
Be sure to check their regular best rate with the rate for park and fly. You may find it just as cheap to use one of the commercial long term parking facilities. If you have a really early flight I can understand the desire to spend the night. When I checked a couple weeks ago, rates were around $60 extra for extended parking and that is usually only for a week and more for extra days.