Minneapolis is served by the fancy new cheap bus with WiFi. One idea might be to get to the bus (Chicago or someplace closer?) using Southwest or something. Some advanced purchase tickets are really cheap. There's also some train service, again certainly from Chicago but can you do better from MCI?

Driving is generally very good interstates that aren't too crowded. It should be fine except in the winter months when blizzards are a major risk and can lead to long delays.

Sometimes connecting tickets are cheaper. Look especially at connecting through MDW/ORD, MKE, STL, even DEN, rather than just pricing the nonstop. (These are reasonably nearby major airports with generally competitive fares; for instance, I have recently seen DL and UA OW MDW/ORD to MSP price as low as $62 depending on the day, without even much of an advanced purchase requirement.)

Does it make sense to rent a car to drive it one way? Have you thought about the "driveway" scheme where you drive someone's private car that otherwise would be shipped to Minneapolis? This might work at a time when the sunbirds are migrating north and don't want to drive themselves.