What did the passengers on the mayflower called themselves?

The pilgrims! They were all of a particular radical denomination of Puritanism called "Separatism," which contrasted with another branch called "Reformism" in that Separatists believed the Church of England was incapable of being reformed to be more in line with Puritanical teachings. Many, including the pilgrims, set out from England in the hopes of establishing a new Puritanical society in America.