Probably her earliest name she ever took, even before she was born, she was Helen and Richards daughter.
he remembers the warm feeling of being held by them, their joy in all her achievements, from first words, to first steps, to first book she ever read.
She remembers the first time the connection of Daughter was tested.
One afternoon after reading Matilda for the umpteenth time, a young Hermione furrowed her brows, and focused on calling over one of her books from across the room, like Matilda in her book had done. To her, and her mother's surprise, the book sluggishly moved off her small bookshelf and wobbled it's way to her. That was the first show of her magic.
So when many years later, a prim and tartan clad professor came to her parents door, Hermione began to worry for her connection to her parents.
Her parents though, were supportive and steadfast to make sure their daughter could enjoy every opportunity she could.
Through her Hogwarts years, she was still Helen and Richards daughter, but in a way she also became Molly and Arthur's daughter, which helped her transition into the magical world.
'Daughter' was also the first name she lost, as she sent her parents so a safer place, as not her parents, but just another childless couple. Later that night, she mourned the passing of the name, and it's connections.