Well, she'd probably start setting up study groups in advance. Since she's so studious, she could pick the best study spots in the common room or the library. And she could also start making friends with some of the older students who are also early - birds. This would expand her network at Hogwarts and give her different perspectives on magic from students who have been there longer.
She could meet some of the Hogwarts ghosts privately. Maybe the Bloody Baron would share some secrets about the castle's history that are not widely known. This would add to her knowledge of Hogwarts and might help her in solving some of the mysteries that pop up during her years at school.
She might have more time to explore the library on her own. Without the distraction of other students at first, she could discover some really rare and powerful magic books. This could give her a head start in her studies and she might even come across some ancient spells that could be useful later in her adventures at Hogwarts.
Hermione could also get some extra study time in the library. With no other students around, she would have access to all the books she wants without any distractions. Maybe she would find some really rare and powerful spells or magical knowledge that would help her throughout her time at Hogwarts.
It could strengthen her relationship with the teachers. Since she is there early, she can make a good impression on them right from the start. They might see her eagerness to learn and give her special privileges or tasks, which would set her apart from other students.
One reason could be for an advanced placement test. The Hogwarts professors might have recognized her exceptional magical abilities from her pre - Hogwarts education and wanted to assess her early. So they invited her to come early to take some special tests to see if she could skip some basic courses.
The students would surely miss her. She's a role model for many, especially the younger students who look up to her intelligence and determination. Without her, the study groups and academic competitions might not be as intense or successful.
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 this fanfiction, Aang might bring his airbending skills to Hogwarts. He could use his bending to create interesting effects during Quidditch matches, like using air currents to speed up the broomsticks. He may also interact with the Hogwarts students in unique ways, perhaps teaching them a bit about his culture from the Avatar world.
Well, he might show off some really old - fashioned but extremely powerful magic that would shock everyone at Hogwarts. Maybe he'd befriend Harry Potter and teach him some ancient spells that could defeat Voldemort in a whole new way.
In this fanfiction, Dudley might face a lot of challenges. He's used to a different world than Hogwarts. He could struggle with magic classes as he has no prior knowledge. Maybe he'd be bullied at first for being a Muggle - born in the eyes of some students. Also, his relationship with Harry could change in a new and interesting way.
There's a chance she meets some old Death Eaters who are lurking around Spinners End. This would put her in a very dangerous situation, but it would also make for an exciting plot. She would have to use her wits and magic skills to outsmart them. Or perhaps she meets some of Snape's old school friends who can provide insights into his character and motives.
Well, perhaps in 'harry goes to hogwarts later fanfiction', he could form new friendships. He might meet students from different backgrounds or even different magical cultures, which would bring a whole new dynamic to his Hogwarts experience. He could also discover new magical abilities or spells that were previously unknown to him, which would help him in his adventures at Hogwarts.