At MSP, I've seen people sleeping on the balcony overlooking the check-in lines. This is landside--outside of security. Take the escalator or elevator one level up from the departures level.

There's no hotel in either airport terminal; the closest hotels are around the Mall of America, the first stop south of the two airport terminal stations on the light rail line, although most of the twenty or so airport hotels offer shuttle service as well as direct phone lines in the baggage claim area. Some of these properties frequently have cheap rooms.

I'm not sure whether the two MSP terminal stay open all night when there's no serious problem like a blizzard. Few planes arrive or depart late in the evening, although there are some very early departures.