These fanfics exist because they add a more human and relatable aspect to Hermione's character. Periods are a normal part of life for many, and showing Hermione going through it makes her more real and three - dimensional.
One popular 'Hermione gets her period' fanfiction might be 'Growing Up at Hogwarts'. It focuses on how Hermione deals with this new aspect of growing up while also handling her studies and friendships at Hogwarts. Another could be 'The Monthly Changes' which shows how her friends, like Harry and Ron, react when they find out and how they support her.
A fanfic named 'When Hermione Got Her First Period at Hogwarts' could be really interesting. It could start with her initial confusion and embarrassment, as she's likely not been fully prepared for it in the magical context. It might then show how she turns to female professors or older female students for advice, and how this experience makes her grow as a character.
In fanfiction, it could show Hermione's friends' reactions. Harry and Ron might be completely clueless at first. Ron could make some awkward and funny remarks out of ignorance, while Harry would be more empathetic but still a bit unsure of how to help. Hermione would then have to educate them a bit about it.
Obviously Caroline herself is the main character. Her mother is likely to be in it as well, as mothers often play an important role in a girl's first period experience, like giving advice and support.
It could be about Caroline's initial shock and confusion when she gets her period for the first time. Maybe she's at school and has to deal with the embarrassment of not being fully prepared.
Well, it could be about Arya's confusion and discovery when she first gets her period. Maybe it shows how she copes with the physical and emotional changes, like feeling a bit scared or embarrassed at first. And how the people around her, like her mother or sister, help her understand what's going on.
It could be about a character named America going through a series of events during a special selection process. Maybe it's a competition where she has to prove herself in different ways.
Well, it could be about Aria dealing with the new experience while also handling the drama in PLL. Maybe she tries to hide it from the other girls at first because she's embarrassed, and then as the story progresses, they find out and support her.