I resemble that quest - done it at SEA LAX SJC ATL and SLC but not here at MSP.

Planning on camping out in the terminal - try and do it on the TSA security side where there maybe a 24 hour coffee shop; fast food venue open or snack machines.
The usual gate furniture and a carpeted nook to snooze horizontal on the floor not much else to be found.
If planning on doing this DO NOT LEAVE security you can not get back in until ticketed the next day.
I have no idea if the airport police clear the terminal at night forcing everyone out !
If able to stay inside the TSA zone you will have the noise of maintenance vacuum and floor scrubbing machines and an occasional announcement on the PA.
Near moving walkways and automated trams you will have that to lull you to restless sleep !
Outside of TSA security -water restrooms and perhaps a snack machine may be found very little in creative sleeping arrangement.
Member of the military or government corps there is a USO Club on the north end of the Mezzanine deck above the ticket counters.

In front of the terminal are the taxi stands and hotel shuttles.
The high end brand name hotels will not have much to offer in rates unless you already have an affinity with them.
The low end hotels may not have a shuttle and a taxi ride of $15 or so will kill any reduced rate.
A thought call the/a hotel and if arriving after midnight inquire if they have a half day rate. They have sold all the rooms that they
expected to sell and any unfilled ones are going to stay that way UNLESS they offer a deal. Most of the airport hotels are about a
15-20 minute ride from the airport - if your next day departure is in the early AM - not terribly practical to go through these motions.
3-5 hours sleep may not be very productive with jumping through hoops out and back through security.

You did not note what your next day plans were or what time of the night you were planning to get stuck.

Since you are part of the airline group - inquire if there is pilot or flight attendant lounge - all they can say is yes or no and it is not for your use.

Thinking of the Sky Club - Closed during the late overnight hours.

A lark take the Light Rail downtown Mpls or even to StP - you can ride for 2.5 hours on one fare round trip if timed right.

Good Luck - If you try this late next year a hotel is planned in the airport - but don't wait up for it ! LOL!