That isn't really a grammatically correct or common-use phrase. The meaning itself is whatever the singer/song-writer interpreted it as. I would guess she means that she will "cry out the tears", to basically cry until there are no more tears to shed (from looking at the rest of the lyrics). "Cry me out" I would think would be similar to the way you would say "ride it out"; to refer to putting up with it until the end, but I could be wrong, that would be my guess. The phrase itself is not something you will likely hear outside of a song, it's kind of "artistic", not common English.

I agree it's not a typical phrasing, but it's fairly easy to understand. "Cry me out" roughly means "cry over the memory of me, until it's out of your system". In short, "get over me".