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.
One possible plot could be Hermione being selected for an exchange program to Durmstrang. There, she might face challenges in adjusting to their different teaching methods which are more focused on the Dark Arts in a controlled way compared to Hogwarts. She could also encounter new friendships and rivalries. For example, she might become friends with some students who are secretly opposed to the more extreme elements in Durmstrang's curriculum.
She could meet Bella Swan. They might start off as a bit of an odd couple, with Bella's more reserved nature compared to Hermione's boldness. But they could learn a lot from each other.
She could meet the headmistress of St Trinians. That would be an interesting encounter as Hermione tries to understand the school's rules and the headmistress tries to figure out this strange, book - smart girl. Also, she might meet the troublemakers of the school, those girls who are always getting into scrapes and might see Hermione as either a threat or someone to rope into their schemes.
Perhaps she'll meet older, more experienced students who can act as mentors. These students could introduce her to secrets of the new school that are not easily accessible to newcomers. They might also help her with any homesickness she has for Hogwarts.
Voldemort's remaining Death Eaters could be behind it. They might see Hermione as a threat to their attempts to regain power in the wizarding world.
Madame Maxime, the headmistress of Beauxbatons, would surely play an important role. She might be initially skeptical of Hermione but later come to respect her. Also, there could be a young male wizard at Beauxbatons who is intrigued by Hermione's intelligence and Hogwarts - style magic. He could try to learn from her and in the process, they become friends and face challenges together in the school.
One possible plot could be Hermione feeling out of place at Hogwarts due to some unforeseen circumstances. Maybe she has a falling out with some of her friends or feels that the Hogwarts curriculum isn't challenging enough. So she decides to transfer to Durmstrang. At Durmstrang, she might face a lot of prejudice as she's a Muggle - born witch. But she could also make new friends and discover new magical techniques that are different from what she learned at Hogwarts.
She could interact with the founders of Hogwarts. Imagine Hermione having a conversation with Rowena Ravenclaw about knowledge and intelligence. It would be a really cool scene as they could exchange ideas about magic and education.
Hermione might also come across some magical creatures that are guarding certain vaults in Gringotts. For example, a dragon that she has to find a way to bypass without getting burned. These encounters would add more excitement and danger to her visit to the wizarding bank.
Maybe it's a new dark wizard or witch who has infiltrated the Ministry. They manage to manipulate the situation so that Hermione is blamed for their own misdeeds and sent to Azkaban. Their goal could be to cause chaos in the wizarding world and Hermione was the only one who could stop them.