Well, I certainly remember the Battle of Midway. But I must admit that having stopped over in Midway Airport numerous times on the way to somewhere else, I just assumed it was midway on many trips. I would guess that it might be the biggest stopover airport in the country.
Chicago also used to be the biggest railroad hub.
I always thought it was named for the same Midway as “The Monsters of the Midway,” the 1940-41 Chicago Bears. They got their name from a park in a nearby part of town. (Actually, the story is more complicated, involving the University of Chicago football team and the 1892 Columbia Exposition, but that’s a digression.)