Maybe Dumbledore steps in at the last moment and takes Harry away from Vernon's house. He could use his powerful magic to protect Harry and remove him from that hostile environment.
It could be Sirius Black. Sirius is Harry's godfather and he is very protective of Harry. He might break out of his confinement (if he is still in that situation in the fan fiction) and rush to Vernon's house. Sirius has the skills and the determination to face Vernon and save Harry. He could transform into his Animagus form, a large black dog, and intimidate Vernon into letting Harry go.
In this fan fiction scenario, Snape might use his knowledge of the Muggle world and Vernon's weaknesses. Snape could have found out some information about Vernon's business affairs or personal secrets. Then he blackmails Vernon into leaving Harry alone. By doing this, he effectively saves Harry from Vernon without having to use too much magic and draw unwanted attention.
It could be that Snape wants to spite Vernon. Vernon represents everything that Snape despises about Muggles who are cruel and unaccepting. By saving Harry from Vernon, Snape is showing his superiority over Vernon and also his power as a wizard. And of course, there's still that underlying reason of his love for Lily which makes him want to protect her son.
In some Snarry fan fictions, Severus Snape himself might have saved Harry from the stalker. Snape, with his powerful magic and his new - found, albeit complex, feelings for Harry in the Snarry context, could have detected the stalker's presence and used his skills to fend off the threat. He has the knowledge and experience to deal with such dangerous situations, so he might have stepped in to protect Harry when he realized the danger Harry was in.
One common way is through the power of friendship. Harry's friends, like Ron and Hermione, might be vigilant and notice the stalker's actions. They could then use their combined magic and wits to outsmart the stalker and save Harry.
Perhaps Dudley finally stands up to his father's cruel treatment of Harry. Maybe he realizes that Vernon's behavior towards Harry is wrong and uses his own physical strength or some clever ruse to get Harry out of a difficult situation Vernon has put him in.
If Snape witnesses Vernon abusing Harry in fan fiction, he might use it as an opportunity to gain some leverage over Harry. Snape could offer to keep quiet about the abuse in exchange for Harry doing certain things for him, like getting good grades in Potions. However, this is a rather cynical view and goes against Snape's more complex character who deep down has some good in him.