Quote Originally Posted by darkTifa
I thought that Advent meant: The coming of something important.
Yes, an advent is the coming of a long anticipated arrival, especially when in reference to something important. Also, the Advent of Christ is the name given to the Second Coming of Jesus. Either of these could be the source of the name in the film. Think of it as either "The Children of the Long Anticipated Arrival," or "The Children of the Second Coming."

Seeing as how (SPOILER)JENOVA/Sephiroth returns in this film, I'm going to assume the name was taken with the "Second Coming" meaning in mind.