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.
Perhaps Harry would find a special wand. Not just any wand, but one that was crafted specifically for the Potters. This wand could have unique properties that are different from the normal wands. It could be more powerful in certain types of magic that were characteristic of the Potter family, like a particular type of defensive magic.
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.
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.
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.
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.
In this fanfiction, his glasses could have a disorder where they constantly fog up, no matter the temperature or humidity. It would be really annoying for Harry. He might be in the middle of a battle with Death Eaters or trying to solve a mystery in Hogwarts, and he has to keep wiping his glasses to see clearly.
One possible reason could be that the twins might have some special abilities or circumstances that draw the Potters' attention more. Maybe they are more outgoing and seem to need more immediate care or guidance. Another reason could be that there is some sort of miscommunication or misunderstanding within the family. For example, the Potters might think Harry is more self - sufficient when in fact he also needs their love and attention.
Perhaps there's a plot against Harry by some dark forces, and the Potters believe that only they can protect him properly. They might think that the people who have been taking care of Harry so far are not equipped to handle the new and more dangerous threats. Also, it could be that the afterlife in the wizarding world has some rules or opportunities that allow them to try and get Harry back into their family fold. They could have been given a second chance, so to speak, and are eager to take it.
There's no real reason for the Potters to give away fanfic. Fanfic is a fan - created content, and the Potters are fictional characters. They don't exist in the real world where fanfic is created and shared.