There's a book titled 20-in-10: Linking Music and Literacy with Twenty, Ten-minute Mini-lessons ... by Luana K. Mitten, Cathy Fink, and Marcy Marxer. The book claims that Lee Hays of the Weavers invented the term "Zipper Songs."

I don't know if I'd agree that Hays invented the term "zipper song," but he certainly used the term. Take a look at Lonesome Traveler: The Life of Lee Hays, By Doris Willens (page 57):

A "zipper song" is a simple folk tune built on repeated lines and, in Lee's words, "so constructed that you have to zip in only a word or two to make an entirely new verse. 'Roll the Union On' is the best example, with its structure allowing you to roll the union over anything you want to roll over, but there are many more...
"now these are obviously songs that don't call for much brain-work. They are songs you can sing three seconds after you've heard the first line. And, because they are very rhythmic and full of bounce, they're inviting - as a matter of fact, that's just what they are, because they derive from the invitational hymns of the old camp meetings.
"'Join the Union' is 'Come to Jesus.' 'Roll the Union On' is 'Roll the Chariot On.' So today the same music invites people to join churches and unions.

I guess one could say "We Shall Overcome" is a zipper song, as are a number of labor and civil rights songs. "If I Had a Hammer" is a good zipper song, although it doesn't really open itself to new verses made up on the spot.

Oh, the Unitarians have a very nice page on Zipper songs here (click).

-Joe-

Oh, and I think we need a link to Rita Hayworth's memorable performance of "Zip." Too bad the voice was dubbed. I understand that Hayworth was a pretty good singer, but was never given the chance to sing in her movies. Here's the link:

https://www.youtube.com/watch?v=gYZJfcyQ3oM