Its name is taken from the Babel fish, a fictional device used for instantaneous language translation in the book Hitchhiker's Guide to the Galaxy. People (or other sentinent beings) place the fish in one ear, where its nutrition process converts sound waves into brain waves, leading to instant language translation.
The name Babel, in turn, is taken from a tower in the Book of Genesis in the Old Testament of the Christian Bible. Built by the descendants of Noah, the tower was intended to reach to Heaven itself, but the Bible says God punished this act of hubris by destroying the tower and confusing the builders' language so they could no longer understand each another’s speech, leading to the welter of languages in existence today.
Several services have been implemented, such as Lost in Translation, that use the Babel Fish service to translate back and forth between English and another language, often with humorous or strange results.