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.
Hermione could start to compare the two schools more. Maybe she'll notice that Durmstrang has a more competitive environment, which could make her appreciate the more collaborative nature of Hogwarts. Or she might see that Durmstrang has some advanced magical techniques in certain areas that Hogwarts lacks, and this could change her view on what a 'good' magical school should be.
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.
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.
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 interesting aspect in fanfiction about Hermione Granger's Muggle friends could be how they react to the wizarding world. Maybe they are initially shocked but then become fascinated by magic.
In fanfiction, Hermione's Muggle friends are sometimes depicted as being very accepting of her magical life. They don't question it too much and are just happy to be part of her world in a way.
Harry would probably be very supportive. He knows what it's like to grow up without parents, and he would see Hermione's decision as a very kind and noble one. He might offer to help in any way he can, like teaching the child some of his own skills in magic or defense against the dark arts.
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.
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.