In the fan fiction, Severus could start to develop a sense of responsibility towards Harry. At first, he may have saved Harry out of a sense of duty to Lily Potter, but as time goes on, he realizes that Harry has his own worth. This could make him more open to forming a bond with Harry, which is a huge change for his character who has been so closed - off for so long. He may also start to question some of his long - held beliefs about the Gryffindors, especially as he sees Harry's courage in the face of his difficult upbringing at the Dursleys.
Severus might start off as just fulfilling a sense of duty. He has always been a complex character. But as he saves Harry from the Dursleys, he begins to see Harry in a new light. He realizes that Harry is not just James Potter's son but an individual with his own struggles. This could lead to him becoming more empathetic towards Harry, gradually shedding some of his cold exterior.
His character could also develop in terms of loyalty. At first, he might be rescuing Harry for some self - serving reasons, like to gain favor with Dumbledore or to stop a threat to the wizarding world that he discovers is related to the Dursleys. But as he spends time with Harry, he could develop a genuine loyalty towards the boy, which is a big shift from his previous animosity.
Obviously, Severus Snape and Harry Potter are the main characters. Severus is the one doing the saving, and Harry is the one being saved from the Dursleys. But there could also be other characters involved. For example, Dumbledore might make an appearance. He could either support Severus's actions or be a bit wary of them. Petunia Dursley could also be a significant character as she is part of the reason Harry needs to be saved from that household.
Maybe Severus discovers some dark magic plot that involves the Dursleys harming Harry in a more serious way. So, he intervenes just in time. He could use his skills in potions and magic to subdue the Dursleys if they resist, and then take Harry to a safe place, like his own secret hideout or a friendly wizard's home.
It could be Voldemort. He might have influenced the Dursleys in some way to harm Harry, and Severus has to save Harry from Voldemort's long - reaching plans. For example, Voldemort could have placed a curse on the Dursleys' house that would eventually harm Harry, and Severus discovers it just in time.
In some fan fictions, Severus might save Harry from the Dursleys because he realizes that Harry is the key to defeating Voldemort in the long run. He might see that the Dursleys' treatment is making Harry weak and less likely to be able to face Voldemort. So, he intervenes, maybe by intimidating the Dursleys into treating Harry better and providing him with proper education about the wizarding world. He could also give Harry some potions to boost his health and magical abilities that have been stunted by the Dursleys' neglect.
Well, perhaps in the fan fiction, Severus saves Harry from the Dursleys but then has to deal with the consequences of going against Dumbledore's plan. Dumbledore might have had a reason for leaving Harry with the Dursleys, and Severus' actions could set off a chain of events that changes the whole wizarding world. Also, Severus could find out that the Dursleys were part of a much larger conspiracy against the wizarding world, and by saving Harry, he uncovers this plot.
He might use his skills in Legilimency. By probing the minds of Voldemort's followers or looking into the general 'aura' of the area where he suspects Harry to be. This way, he can get a sense of Harry's location even if it's well - hidden.
In these fanfictions, Severus often develops a more nurturing side. Since he has Harry as a pet, he has to take on a caretaker role, which is different from his usual strict and aloof self. He might become more patient as he guides Harry through various situations.
Severus' character often develops from a cold and aloof figure to a more caring and nurturing one. He starts to open up his heart as he takes on the role of raising Harry. His sense of protectiveness grows, and he becomes more patient, which is a big change from his usual self in the original series.