In some fanfictions, Hermione might be so focused on following the rules that she opposes Harry's more risky but potentially effective plans. Harry, being the more action - oriented one at times, would tell her off for not being more flexible and trusting his instincts.
One reason could be Hermione's excessive worrying. She often frets over small details and Harry might get tired of it and tell her off. For example, if she keeps nagging about the proper way to perform a spell when they are in a hurry during a dangerous situation.
Maybe Hermione chooses Ron over Harry in the fanfiction. Since Harry has deep feelings for Hermione, this would be a crushing blow to him.
It might be related to Hermione's research. She may have read about some unique magical phenomenon in Australia that she wants to study first - hand. And Harry, being her loyal friend, goes along with her. For example, there could be a special type of plant with healing properties that only grows in certain parts of Australia.
It might be because of a misunderstanding. In some fanfics, a false rumor could spread about Harry. Ron and Hermione, believing it to be true, might decide to distance themselves from him. Say, there's a rumor that Harry is involved with dark magic in some way, and they, being the 'good guys', think they need to stay away.
They could be on a mission to exchange magical knowledge. Hogwarts has its own set of spells and magical techniques, and so does the American wizarding world. They might go to America to learn new things and share their knowledge as well.
She might wear his clothes because of necessity. For example, if her own clothes got damaged or lost during an adventure. Harry's clothes would be readily available and a practical solution.
It could be for safety reasons. If there's a threat at Hogwarts or in the wizarding world in general, Hermione's house might be a safe haven. Her family might have some protective spells around the house that Harry can take advantage of. Also, Hermione's parents are Muggles, so it might be a place where the dark forces wouldn't expect them to hide.
Some might write it to explore a different side of the characters. Maybe they want to see how the story would change if there was conflict between Hermione and Harry instead of their usual friendship.
In fanfiction, Hermione might choose Harry over Ron because Harry often has a more heroic image. He is the Boy Who Lived, and his adventures are often more central to the main plot. Hermione could be attracted to his leadership qualities and the fact that he always seems to be at the heart of the action.
Maybe she's desperate for some knowledge that she thinks Harry has in those books. For example, if the books are related to some dark magic or ancient spells that she wants to study for the greater good, she might be tempted to take them without asking.