When brothers Eric and Andrew Dayton ordered 150 hats boldly knitted with the word “North” to sell in their trendy Minneapolis store, Askov Finlayson, the hats sold out in four days. That was last winter; the hats have since been impossible to keep in stock.

“North” has a special meaning in Minnesota these days, and it is gradually gaining a stronger following. Though most Americans consider the state part of the Midwest, a number of local influencers are proposing to redefine Minnesota as a region that the U.S., officially...