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.
Harry might become withdrawn and introverted. Feeling unloved and unwanted, he could start to keep to himself more, spending time alone in his room or exploring the grounds of their home alone, trying to make sense of why his parents seem to favor the twins.
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.
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.
One possible motive could be that the Potters discovered some dark plot by the Malfoys involving Harry. Maybe they found out that the Malfoys were planning to indoctrinate Harry into their pure - blood supremacist ideology against his will. So, to protect Harry from that, they kidnapped him.
The twin could change the plot by being more interested in academic magic than Harry. This might lead to him uncovering secrets in the library that Harry misses, which could be crucial in the fight against Voldemort.
One common theme is loyalty. The twin brother is always loyal to Harry, standing by him no matter what. For example, in many of these fanfics, when Harry is wrongly accused, the twin brother is the first to defend him.
I've also heard of 'Twin Guardianship'. In this fanfic, the twin brother is more attuned to the secrets of Hogwarts. He discovers hidden passages and rooms that can be used as safe havens for Harry. He also forms alliances with some of the more mysterious creatures in the school grounds to keep an eye on Harry from all angles.
Well, a cool plotline could be that the twin brother has a relationship with a character that Harry doesn't really interact with much in the original series. Say, he becomes friends with a Slytherin who has a different view on the wizarding world. This friendship could bridge the gap between the houses in a way that Harry's relationships in the original story don't. It could also lead to the twin brother having a different perspective on the ongoing conflict between good and evil in the wizarding world.
Obviously, Harry and his twin brother are the central characters. But other popular characters can include Hermione and Ron. They might interact with the twin brother in different ways compared to how they interact with Harry. For example, Hermione might help the twin with his studies more because he has a different learning style. Dumbledore is also often a key figure. He could have different plans for the twin brother, like entrusting him with a special mission that is different from Harry's.