In this kind of fanfiction, perhaps Harry is rescued by the Order of the Phoenix earlier than in the original story. They might have realized the Dursleys' mistreatment sooner. Then Harry could start his magical training earlier, away from the hostile environment of the Dursleys' home. He may also form closer bonds with other members of the Order, like Sirius, who could play a more active role in his upbringing.
Hermione could be the rescuer. Maybe she discovers through her research that Harry is in a bad situation at the Dursleys' and decides to take action. She uses her intelligence and magic skills to break into the Dursleys' house and get Harry out. Then they would go on an adventure together, and Hermione would be like a mentor to Harry in a way, teaching him things she has learned from books.
It could be Dumbledore. He might have realized that the Dursleys were not the best place for Harry after all and decided to take matters into his own hands, using his powerful magic to rescue baby Harry.
Perhaps Harry is rescued by an unexpected character, like a centaur who has been secretly watching over him. The centaur arrives at the Dursleys' on a moonlit night, and using its natural magic and knowledge of the stars, it convinces Harry to come with it to a safe place in the Forbidden Forest where he can be protected and trained.
One possible scenario could be that a powerful wizard or witch, perhaps someone who knew James and Lily Potter well, comes to rescue baby Harry. They might use magic to bypass the Dursleys' defenses and take Harry away to a safer place, like a hidden wizarding community.
There are fanfictions where a powerful wizard or witch, not part of the main Hogwarts faculty, discovers Harry's situation. Maybe an old hermit wizard who lives on the outskirts of the wizarding world. This wizard uses his magic to create a diversion at the Dursleys' place and then whisks Harry away to his own hidden cottage where he teaches Harry different kinds of magic not taught at Hogwarts.
Perhaps Snape uses magic to create a distraction at the Dursleys' place. He could make strange noises or apparitions that scare the Dursleys, and while they are in a panic, he grabs Harry and disapparates. This would be a quick and efficient way for him 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.
Dumbledore might play a role. He could be the one who encourages Harry to stand up to the Dursleys. He might have some hidden knowledge about the relationship between the wizarding world and the Dursleys that could help Harry. Also, Hagrid could be involved. He could provide some muscle if things get physical. He might also offer some emotional support to Harry, as he's always been like a big, friendly giant to him.
In this fanfiction scenario, Bellatrix saving Harry from the Dursleys is quite an unexpected twist. It could be that Bellatrix has some ulterior motive. Maybe she wants to use Harry for her own dark purposes against Voldemort. She might see Harry as a valuable pawn in the power struggle within the Death Eater ranks.