Maybe he wants to explore his mother's side of the family more. By taking on the name Evans Potter, he could be trying to connect with his mother's heritage and the magic that might be associated with it. It could also be a way to honor his mother in a more direct way, rather than just being known as a Potter.
One possible motive could be to distance himself from the Potter name which has a huge target on it. Since he's constantly in danger because of Voldemort's vendetta against his family, becoming Evans Potter might throw off his enemies.
In some fanfictions, the motive could be to create a new identity for himself in the wizarding world. Harry might be tired of the expectations and the fame that comes with the Potter name. As Evans Potter, he can start anew, make different friends, and perhaps even study magic in a different way without the constant pressure of living up to the Potter legacy.
One possible motive could be extreme revenge. If the Muggle relative, like the Dursleys, had been particularly cruel to Harry over the years, some fanfiction writers might explore the idea of him snapping and seeking ultimate revenge. Another could be a dark magic influence. Maybe in the fanfic, Harry is exposed to some dark magic that corrupts his mind and makes him view his Muggle relative as an obstacle to be removed.
Well, it could be for the sake of exploring a different kind of magic or a different set of rules in the wizarding world. Perhaps in this fanfiction's version, the parents were under some sort of dark magic influence that made them turn on Harry. Or it could be a psychological exploration. The writer might be interested in delving into the minds of Harry's parents and seeing what could make them do such a heinous act. It could also be just a very different take on the concept of family and love in the 'Harry Potter' universe.
One possible motive could be misinformation. If Harry is fed false information about Sirius being a traitor or a danger, he might act on it. For instance, if a well - disguised Death Eater manages to convince Harry that Sirius has joined Voldemort, Harry, in his haste to protect others, could attack Sirius.
In a fanfiction, Dumbledore might steal out of desperation. Suppose there was a situation where a curse was about to be unleashed on Hogwarts, and the only way to stop it was by stealing a particular ingredient for an antidote from a place where it was not supposed to be taken. Dumbledore, seeing no other option, decides to steal it. His motive here is to save the school and its students, even if it means breaking the rules.
Lucius may also have a moment of moral awakening. Despite his past actions, he could see the value in Harry's life and the good that he represents. Maybe he has a change of heart after seeing the destruction caused by Voldemort's actions and decides to do the right thing for once by saving Harry.
One possible motive could be extreme revenge. After years of being mistreated, locked in a cupboard, and generally being made to feel unwanted, Harry might snap in fanfiction. Another could be a sense of self - preservation if the Dursleys were somehow involved in a plot against Harry's life in the fan - created plot.
One popular fanfiction with this concept might be 'The Evans Potter Chronicles' which delves into how Harry's decision to change his name impacts his relationships within the wizarding world. It shows how his friends react and how it changes the power dynamics. Another could be 'A New Identity: Evans Potter' which focuses more on his internal struggle and the reasons behind wanting the name change, like escaping the Potter legacy's burden.
One possible motive could be a misunderstanding. Maybe Snape is framed for something really bad, like a dark magic plot against Hogwarts. Harry, in his haste to protect the school and his friends, believes Snape is guilty and takes drastic action. Another reason could be if Snape accidentally endangers someone Harry loves dearly, like Ginny or Ron, and Harry's emotions get the better of him in the heat of the moment.
One possible motive could be that the 'mountain' in the fanfiction represents a great evil or obstacle that is threatening the wizarding world. Harry, being the hero, feels it's his duty to eliminate it. For example, if the mountain is a lair for dark creatures that are attacking Hogwarts students.
Maybe Sirius wants to train Harry himself. In fanfiction, he could believe that he has a better understanding of the Dark Arts and how to fight Voldemort compared to the professors at Hogwarts. So, he kidnaps Harry to start his own special training regime away from the school.