I toured through across Northern MN and down the Mississippi River a couple of seasons ago. If you get a chance to go through the small town of Dalbo, MN, look up Donn Olson's bunkhouse. Adventure Cycling has his information--it's on their Northern Tier route. Donn knows how to do it right.
Most of my free camping in the state was only quasi-legal. I never trespassed on private property, but I stayed in public parks where camping wasn't allowed and it was easy to hide.
It's easy enough to bathe for free everyday in that area, too. There are plenty of public park restrooms where a sponge bath is easy.
One of your major expenses may be milkshakes, traveling with a young boy. Have a great time.