Yes, they might. Just like any other group of people, those who identify as Blacks or are part of the Potter family name (if we assume there are real people with that name) could be fans of the Harry Potter series and thus read fanfiction. Fanfiction offers a wide range of interpretations and extensions of the original story, and anyone with an interest in the magical world of Harry Potter could be drawn to it.
One reason for Blacks to read Harry Potter fanfiction could be the representation (or lack thereof) in the original series. Fanfiction can sometimes address issues like better representation of diverse characters. For Potters, it could be the curiosity of seeing their name in different fictional scenarios within the Harry Potter world. Also, the popularity of the series means there is a vast amount of fanfiction available, and both Blacks and Potters could be attracted to the variety of stories, whether it's about new magical creatures, different relationships between characters, or alternative endings to the main story.
Well, he could discover new relationships his parents had. Maybe his mother Lily had a close friendship with a character we don't know much about in the official story. Fanfiction often delves into uncharted territories of a character's life. So, Harry could find out about the Potters' early days at Hogwarts, like their first encounters with certain teachers or how they formed their own group of friends.
One interesting story could be about the Potter family's daily life before Harry goes to Hogwarts. Maybe how James and Lily Potter raised Harry in his early years, their family traditions, and the magic they used around the house.
I'm not sure if the Potters (as in the characters in the Harry Potter series) would give away Harry Potter fanfic. In the official lore, there's no indication of them doing so. Fanfic is more of a creation by fans in the real world rather than something the fictional characters would be involved with.
Bellatrix Lestrange could also be a main character in some versions of this fanfiction. Despite her being on the 'dark side', her complex relationship with the Black family and her powerful magic could make for some intense storylines. Harry might have to deal with her in some way, either trying to redeem her or facing her as an enemy. And of course, Remus Lupin could be involved too, as he was friends with Sirius and has a connection to Harry. His werewolf nature might add another layer of complexity to the story.
One possible plotline could be Harry learning about the Black family secrets that were long hidden from him. Maybe he discovers some ancient magic that was passed down within the Black family and uses it to his advantage in his fight against Voldemort.
Well, in such a fanfiction, a possible plot point is the way magic is taught to Harry at home by the Potters. It could be a more personalized and in - depth learning compared to what he would have received at Hogwarts. Also, the family's influence might lead to different friendships for Harry. For example, he might be closer to certain pure - blood families that the Potters are associated with from the start, which would change the whole dynamic of his social circle at Hogwarts.
In fanfiction, Harry's understanding of the Potters can expand. He might see his parents as more complex characters. For instance, if a fanfic shows James Potter as having a struggle with a certain type of magic, it makes him more than just the heroic figure Harry knew. This changes Harry's view of his father.
Obviously, Harry Potter himself is a main character. He's at the center of many of these fanfictions, whether it's about his growth within the family or his relationships with other family members.