'Twilight' refers to the vampire's world that Bella entered when she fall in 愛 with Edward. 'New moon' refers to the darkest time of Bella's life when Edward left her, because the night is so dark when it's new moon. 'Eclipse' refers to what Bella used to think that Jacob is her sun but Edward overshadowed him, so he can't win over him. 'Breaking dawn' refers to the transformation of Bella's life from ordinary human to so powerful vampire that shines.
I think I read somewhere that Twilight was on a 一覧 of 'words with atmosphere' and I guess they went with a theme of sun/moon related words. It did sort of mirror Bella's story arc - as the story gets darker the sun/moon phases seem to get darker.