She could make a spell - powered communication device. This would be like a magical version of a smartphone. It could allow wizards to communicate instantly across long distances, even in places where normal communication spells don't work. It would be a great invention considering how often wizards need to communicate in the wizarding world.
She could write about the cultural exchange between the wizarding world and the Shadowhunter world. In her fanfiction, she might introduce the concept of Quidditch to the Shadowhunters. Or perhaps she writes about how Shadowhunter weapons could be used in the wizarding world. These stories would not only be interesting but also show how two different magical worlds could learn from each other.
She could be a maester - like figure. With her vast knowledge, she could advise rulers on various matters, from war strategies (using her strategic thinking from Hogwarts battles) to matters of health and economy. She'd be different from the traditional maesters as she has a more modern - thinking approach.
She could have the power of super - intelligence as a demigod. This would make her even smarter than she already is in the original story, allowing her to understand complex magical theories and ancient spells with ease.
Maybe she leaves because of a conflict. It could be with a teacher who doesn't support her unorthodox ways of learning or a fellow student who makes Hogwarts an unpleasant place for her. In fanfiction, there could be a lot of creative interpretations of such conflicts.
Well, in a Hermione Granger family reunion fanfiction, perhaps Hermione gets to introduce her wizard friends to her Muggle parents in a really fun way. Maybe they have a big dinner where they share magical stories that Hermione's parents are just amazed by.
Well, as a centaur, Hermione would likely gain some centaur - specific magical powers. She could use her enhanced speed and strength to explore areas that were previously difficult for her as a human. And she might communicate with other centaurs in a way that reveals new magical secrets about Hogwarts or the Forbidden Forest.
Well, it could be about Hermione and Crookshanks having some sort of magical adventure together. Maybe they discover a hidden part of Hogwarts that only Crookshanks' cat - like senses could detect. And Hermione uses her vast knowledge to solve the mysteries they encounter there.
The plot could revolve around the older sister being a secret agent of the Order of the Phoenix. She could be involved in dangerous missions while also trying to protect Hermione from getting too involved in the dangerous wizarding politics. For example, she might try to keep Hermione away from Harry Potter's dangerous adventures at first.
One implication could be in her social status. She would be more accepted by some of the pure - blood elitists at Hogwarts. Instead of facing discrimination as a Muggle - born, she might be part of the inner circle. Another aspect is her access to resources. A noble house would likely have a large library or rare magical items, which would enhance her learning even more.
Well, it could symbolize a new beginning. Just like in real life, a new haircut can mark a change in a person. In fanfiction, Hermione getting a haircut might show that she's ready to break free from her old self, perhaps be more daring.