Cotton balls are also used for cleaning around the umbilical cord.

Most hospitals don't require you to take extra on top of bunny rugs. I think they are called bunny rugs, because they traditionally had bunnies on them or it was a well know brand name. Sounds strange but I think it's just one of those things, when something is sooo popular, the name becomes generic.
Muslin is great for warmer climates or very warm hospitals also.