Definitely don't park at the airport!! We only live about 45 minutes from MSP and we often spend the night before our flight at a hotel in the airport area so we can take the shuttle to the airport. We always fly out early in the morning and the convenience of drop-off at the door with luggage is fantastic! It will definitely cost you less to pay for the hotel room than to pay for parking, even at the value lot. And if you're driving in the day before anyway, it's a total no-brainer.
Here's a list of hotels that have shuttles to the airport:
http://www.mspairport.com/GroundTransportation/hotel-shuttles.aspx
We have used the Comfort Inn Eagan a few times and the Super 8 as well. Both served our purpose nicely. I would take some time and look up each hotel that interests you or make some calls to check their policies and figure out which works best for you. The rates do vary a bit and it's worth your time to do some research.
You will want to check with the hotel to see if they have a limit on the number of days you can leave your car at the hotel. The Super 8, if I remember correctly, allows you to leave your car for 7 days with a one-night stay and require a two-night stay for leaving it longer than 7 days. If you're staying the night you return too, that would likely work for you. The Comfort Inn allows you 8 days with the one-night rate and charges $10 per day after that. Still cheaper than parking.
The other thing you need to ask about is the shuttle times. Some offer a range of times (like 5 am to midnight) while some offer a 24 hour one. I think the Super 8 was 24 hours but goes to (don't quote me here) Terminal 1 on the hour and Terminal 2 on the half hour. That required some planning for us when our flight left at 6 am.
If you're not spending the night you return at the hotel, no problem. You just call for the shuttle when you arrive and they will take you back to the hotel to get your car. Sometimes the driver is nice and will take you directly to your car. (Those guys get a nicer tip from me!) Most will tell you to call for the shuttle when you have picked up your luggage, but we usually call when we get to baggage claim so we don't have to wait as long. That's a bit of a gamble if for some reason you have a luggage issue, but most times it will be fine. They will tell you where to go for pick-up.
It's really a nice service that these hotels provide and definite the best option for you. Feel free to throw me any questions you might have. I'm happy to help!