Perhaps when Harry goes to Hermione's house, they end up having a deep conversation about their future in the wizarding world. They could discuss their plans for after Hogwarts, like Hermione's dreams of reforming the Ministry of Magic and Harry's thoughts on teaching Defense Against the Dark Arts at Hogwarts. They might also share some personal feelings they've been hiding from each other.
In the fanfic, Harry might have a study session with Hermione. They could go through some old magic books, looking for answers to a mystery at Hogwarts. Maybe they'll find a new spell or a clue about a hidden part of the school's history.
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.
There could be a group of students who are known for their pranks in the fifth house. Harry would probably cross paths with them and might either get involved in their antics or try to stop them. Another possibility is that he meets a shy student who has some special abilities but is too afraid to show them. Harry could be the one to encourage this student to come out of their shell.
One of her friends could be a girl who is also interested in Muggle Studies, which is not really a popular subject in Durmstrang. They meet in the library while looking for books on the topic and start sharing their views on the Muggle world and magic. Also, there might be a young wizard who is against the more aggressive and dark - leaning teachings of Durmstrang and sees Hermione as an ally in changing the school's culture.
It could be when they need to discuss a crucial plan related to their adventures. Maybe Hermione has some important information or resources that they need to access together.
Well, they could meet younger versions of themselves and their friends. It would be really interesting to see how they interact. They might also try to give some hints or advice to their younger selves, but they have to be careful not to disrupt the timeline too much. And they could discover secrets that were long forgotten in the present time.
Hermione may become more cunning. In Slytherin, she has to adapt to the house's traits. So she might start using her intelligence in more devious ways, like setting up elaborate traps for those who try to bully her, but still within the bounds of good. For example, magically rigging a classroom so that when a bully tries to embarrass her, they end up getting a face full of slime.
One possible scenario could be that they are on a secret mission. Maybe they are looking for some hidden knowledge or a powerful magical artifact that Snape had stashed away in his house. Another scenario might be that Harry is in hiding at Snape's house for some reason, and Hermione comes to check on him and bring him supplies or important news. It could also be that they are reminiscing about Snape after his death and going through his things to understand him better.