Well, it could be a story where Harry stumbles upon the second branch of the Potters. Maybe it's a long - lost part of the family that has some unique magical abilities or secrets. Since it's a fanfic, the author might take creative liberties and introduce some interesting plot twists, like the second branch being involved in a hidden magical society or having a connection to an ancient prophecy that Harry then gets drawn into.
Harry Potter is a given as the main character. In terms of the second branch of the Potters, it could include a variety of characters. There might be a Potter who has been living outside of the main wizarding society and has developed some unique magical skills. There could also be a group of Potters, like siblings or cousins from the second branch. One of them might be a bit of a rebel and another a more traditional Potter. These characters would interact with Harry, and their different personalities and abilities would drive the story forward.
There's a 'Harry meets other Potters' fanfic where they meet in a time - traveling scenario. Harry from different points in time comes together. It's really cool as they compare their different experiences at Hogwarts, like how magic was taught differently in their respective times, and they also exchange spells and magical knowledge.
Sirius Black could also be part of it. He was Harry's godfather. He might have escaped from Azkaban earlier in this fanfic. He teams up with the Potters to rescue Harry. He could be the one who scouts out the Dursleys' house before the actual rescue operation, using his skills as an Animagus to stay hidden.
Another possibility is that Harry could find a hidden diary of his parents. The diary could contain their thoughts, experiences at Hogwarts, and even some secrets about the wizarding world that they were involved in. It would give Harry a more personal connection to his parents and their lives before him.
Harry could find a collection of letters. These letters might be from his parents to their friends or family members. Through these letters, he could learn about their relationships, their hopes and dreams, and get a better understanding of who they were as people. For instance, letters to Sirius or Remus could reveal a lot about their friendship and the plans they had for the future.
You can try looking on fanfiction websites like Archive of Our Own (AO3). It has a large collection of fanfictions, and you can search for 'Harry meets other Potters' specifically. There are also tags and filters you can use to narrow down your search to the exact type of story you're interested in.
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.
It could also be that the Potters are under some sort of magical influence or curse. In the fan fiction world, there could be a dark force at play that makes them see the twins as more important. This force might be manipulating their emotions and perception, leading to the neglect of Harry.
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.
In this fanfic, the Potters could have faked their deaths all along. They come back to rescue Harry when they find out how horribly the Dursleys are treating him. They might bring some magical creatures with them to help. For example, a phoenix that can heal Harry if he has any injuries from the Dursleys. Then they take Harry with them on a magical journey to a hidden wizarding community where they can all be safe and happy.