Why would Harry go to Hermione's house in fanfiction?

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.

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.

One reason could be to search for a particular magical item. Maybe Dumbledore left behind a clue that led them to believe there was something important in one of the vaults at Gringotts.

Maybe Hermione chooses Ron over Harry in the fanfiction. Since Harry has deep feelings for Hermione, this would be a crushing blow to him.

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.

They might go into hiding because there are still some remnants of Voldemort's followers who are seeking revenge on Harry. These followers could be more dangerous than before as they are desperate and hiding in the shadows. Ginny, being Harry's wife, is also in danger, so they choose to go into hiding to avoid any potential attacks.

One reason could be to study rare magical plants. Australia has a diverse range of flora, and some of it might have magical properties that are not found anywhere else. So, he travels there to learn about these plants and their potential uses in potions or other magical practices.

They might go to the Burrow to reconcile. Fanfiction often explores the idea of enemies becoming friends. The Burrow, being a place full of warmth and family, could be a good setting for them to start over. It's a place where they can put aside their differences and get to know each other better without the pressure of Hogwarts politics.

Perhaps he was on a quest for a rare magical ingredient only found in PT.

It might be that Harry has a certain calmness about him in the face of chaos. Hermione, who is often in control of her emotions, might find herself losing it when the situation gets really bad. Harry, with his ability to keep a level head (most of the time), is the one she wants to be near. So she cries on his shoulder as a way to draw strength from him and also to just have someone to share her heavy feelings with.