To me, Billy Pilgrims's name suggests the fact that he is in fact a pilgrim in any number of ways. He is going to visit Tralfamadore, jump back and forth through time, and be a pilgrim in all kinds of things. I am not quite as sure about Billy, but in a way it gives him a rather regular, every day kind of name that allows people to identify with him, to think of him as both a regular guy as well as someone who travels to new places.
The other part of the name that I think could be meaningful is that he is Billy, not Bill or William. Billy allows him to be both old and young at the same time, neither one being specific to a particular age.