Hermione.
That was the second name ever given to her. After her parents love of Shakespeare and their humorous knowledge of geek mythology.
It was a tricky name for a child to pronounce, and there were different versions over the years, from shortenings, butcherings, and when she was in trouble with her mother; Hermione Jean.
She wore it like armour for most of her life. After all, there weren't many children in the playground called "Hermione". It labeled her as an outsider in her first world, and in her second world it cast her as a new being. On a level all of her own.
She hid it for a time, in one of her most trying years, a year of running, camping and hunting. She hid behind a new name, Penelope Clearwater, a safe name that hid her in plain daylight among darkness.
After the War she used her name as a type of currency. She wasn't proud of it, but she achieved her goals that way. She used her name to prove herself right, and to carve a new path for those behind her who also grew up in another world.
After all Hermione Granger was hers and hers alone, and she had fought for her name.