'Twilight' refers to the vampire's world that Bella entered when she fall in Любовь 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.
The book is named "twilight", "New Moon", "Eclipse" and "Breaking Dawn" because it has something to do with transformation of human bodies to Сверхъестественное bodies which usually occur during these times.
I think I read somewhere that Twilight was on a Список 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.