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.
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 the fanfic, Snape might use his knowledge of the Dursleys' house layout. He may enter the house unnoticed, find Harry, and then use a Portkey to quickly get them both out of there. He could also have some sort of charm or potion that he uses to counter any magic the Dursleys might have used to keep Harry there, like a spell - breaking potion. And he may have planned the whole rescue very carefully, choosing a time when the Dursleys are least likely to notice, like in the middle of the night.
Hermione could be one to rescue Harry. She might have discovered some new spell or potion that allows her to bypass the protections around the Dursleys' house and get to Harry. Then she'd use her intelligence and quick thinking to get him out of there safely.
Maybe Draco finally understands that Harry is a victim of circumstances. The Dursleys' treatment of Harry is cruel, and Draco, having grown up in the wizarding world, realizes it's wrong. So he steps in to rescue Harry.
Maybe Percy uses his magic skills to create a distraction. He could conjure up some strange creatures outside the Dursleys' house. The Dursleys, being afraid of all things magical, would be distracted long enough for Percy to sneak in and get Harry out.
It could make Harry more empathetic. Snape is a complex character, and through his rescue, Harry gets to see a different side of him. This could make Harry more understanding of others' hidden motives and pain. He might start to look beyond the surface when dealing with people, which would be a great development for his character as he faces more complex relationships in the wizarding world.
A rogue wizard who has his own ideas about Harry's future could take him. This wizard might think that he can better prepare Harry for the fight against Voldemort than Dumbledore or the Order of the Phoenix.
Dumbledore could rescue Harry. He's always been watching over Harry and would surely sense if the Dursleys had abandoned him. Dumbledore has the power and wisdom to take Harry to a safe place, perhaps a different magical family or a special hideout. He could also start teaching Harry about his past and his parents earlier, which would give Harry a better understanding of his place in the wizarding world.
Perhaps Snape discovers some dark plot involving the Dursleys that could harm Harry more than he initially thought. The Dursleys were known to be cruel to Harry, and Snape might have found out that they were involved with some Death Eaters or dark magic users who planned to use Harry in a harmful way. So, he decides to rescue Harry to prevent this from happening.