Pete Buttigieg is offering a highly unusual proposition: He’s a revolutionary messenger trying to deliver a safe package.

Mr. Buttigieg, the former mayor of South Bend, Ind., is in many ways the man to watch in Tuesday’s New Hampshire primary. He was the surprise force in Iowa’s caucuses last week, though it is hard to say whether he won, tied or came in just behind Sen. Bernie Sanders, given the muddled counting process there.

In...