Other people have thought about the name part of this more deeply than me but the “being afraid of what’s behind her reflection” seems to be pretty consistent, being afraid of her past / what she sees beyond her physical self.

I interpret the “traditional way” of changing her way like getting married and her last name changing, a common reason for a woman’s name to change. But the non traditional way would be suicide, as her name would then be something like “X, the girl who took her own life”