In the fanfic world, there could be a prophecy that Voldemort has newly discovered. This prophecy might state that Harry needs to be in a different environment for something to happen that will benefit Voldemort. So, instead of leaving Harry with the Dursleys, he decides to save him. He might also believe that the Dursleys are a hindrance to whatever grand plan he has for Harry. Maybe their negative influence on Harry is preventing Harry from reaching his full potential which Voldemort wants to harness for his own goals.
It could be that Voldemort has a new plan for Harry. He might think that by saving Harry from the Dursleys, he can gain Harry's trust. Then, he can manipulate Harry into joining him or at least not being an enemy. After all, the Dursleys have been so cruel to Harry. Voldemort might see this as an opportunity to be the 'savior' in Harry's eyes and then use that to his advantage in the long run.
Perhaps he realizes that Harry has potential that can be used for his own purposes rather than just killing him. So he wants to keep Harry close and away from the Dursleys who are just muggles.
Love for Lily. Since Harry is Lily's son, Snape might feel an obligation to protect him. He couldn't save Lily, but he can try to keep her son safe.
One motivation could be self - preservation. Draco might realize that if Voldemort kills Harry, it will lead to an all - out war that could destroy everything he knows. By saving Harry, he hopes to find a more peaceful solution and protect his own family and the wizarding world in a broader sense.
Well, in some fanfic scenarios, Voldemort might have made a deal with someone who has access to the Dursleys' house. For example, he could have bribed or threatened a neighbor or someone who works in the area. Then, when the time is right, he enters the house under the guise of this person's help and rescues Harry. It could also be that Voldemort has some sort of inside knowledge about the protective spells around the Dursleys' place. He finds a loophole in those spells and slips in to save Harry.
In some fan fictions, Voldemort might save Harry from the Dursleys out of a newfound sense of self - interest. Maybe he realizes that Harry has some power or knowledge that he needs, and by rescuing him from the Dursleys, he can gain Harry's trust and access that. For example, if there is a prophecy that only Harry can unlock and Voldemort believes it will help him achieve ultimate power, he would take Harry away from the Dursleys who are mistreating him.
One possible plot could be that Voldemort realizes Harry's importance in a different way. Maybe he sees that Harry has some power or knowledge that could be useful to him in the long run. So, instead of wanting to kill Harry, he decides to rescue him from the Dursleys to gain his trust and manipulate him later. For example, Voldemort might think that Harry has a connection to some ancient magic that only he can access, and by saving him, he hopes to get access to it through Harry.
Perhaps in this fanfic, Harry has a vision of a future where saving Voldemort is the key to saving the entire wizarding world. So, he travels to various dangerous and mysterious places. He might meet some long - lost magical creatures who have the knowledge of how to break the cycle of Voldemort's evil. Harry has to overcome many challenges, like powerful dark spells and dangerous traps along the way.
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 certain fanfic scenarios, Harry's parents may have left behind some enchanted objects. These objects could have special powers. When Harry discovers them, they might activate and create a magical portal or something similar that allows him to leave the Dursleys' place. For instance, a locket that when opened, transports Harry to a safe place where his parents' magic has created a haven for him.
It could be out of loyalty to the wizarding world. Harry is seen as an important figure in the fight against the dark forces. Fleur realizes that Harry's safety is crucial for the future of the wizarding community, so she saves him from the Dursleys who are a threat to his well - being in a different way compared to the dark wizards.
Well, it could be for power. Maybe Voldemort realizes that by saving Harry, he can gain some sort of power over him or use him as a tool in a more elaborate plan.