Snape's character often develops from a cold and aloof figure to a more caring and nurturing one. At first, he might be hesitant and gruff, but as he spends more time with Harry, his protective instincts kick in. He starts to show more patience and understanding, which is a big change from his usual self in the original series.
It could be an orphaned wizard child from a family that was killed by Death Eaters. This would give Snape an added sense of responsibility as he might feel guilty for being associated with the Death Eaters in the past.
In such a fanfiction, one possibility is that Snape would initially struggle with his new role as an adoptive father. He has a complex personality, but over time he might start to show his protective side towards Harry and Hermione. He could teach them advanced magic, especially his own areas of expertise like Potions, in a more personalized way.
In some 'Snape Adopts Harry' fanfictions, there is a plotline where Snape and Harry have to work together to uncover a mystery at Hogwarts. This mystery could be related to the Dark Arts or some hidden secrets of the school. Along the way, their relationship evolves from a reluctant guardian - ward relationship to a more loving father - son bond. Harry might start to understand Snape's past actions and Snape, in turn, learns to open up and be more affectionate towards Harry.
One possible plot could be that Snape, despite his initial resentment towards James Potter (Harry's father), sees the vulnerability in little Harry. He decides to take him in out of a sense of duty to Lily (Harry's mother). Snape then has to adjust his life to care for a child. He teaches Harry potions at an early age and tries to protect him from the darker elements in the wizarding world. Harry grows up with a different perspective on Snape than in the original story.
In this kind of fanfiction, perhaps Severus Snape and Tom (presumably Voldemort in his Tom Riddle form) would have a very different dynamic with Harry. Maybe they would try to shape Harry's magical abilities in their own ways. Snape might use his knowledge of potions and dark magic to train Harry, while Tom could introduce Harry to some of the more complex and dangerous aspects of magic from a different perspective.
Well, it could be a really interesting story. Snape, being the one to adopt Harry, would have to deal with his own complex feelings towards James Potter while raising Harry. He might initially be strict, but gradually, as he sees Harry's own personality develop, he'd grow to have a very deep and loving bond with him. He could also introduce Harry to a different side of the wizarding world, one that is more influenced by his own Slytherin roots.
At first, it would be really rocky. Harry has a history of distrusting Snape, and Hermione might be a bit cautious too. But as time passes, Snape's care for them in his own strict way would start to show. He'd probably be tough on them about their studies, but they'd see it's because he wants them to succeed.
At first, it's likely to be tense. Harry has years of seeing Snape as a bully. But as they start living together, they begin to understand each other. Snape might show his more vulnerable side to Harry, like his memories of Harry's mother. Harry, in turn, starts to respect Snape's knowledge and sacrifice. Eventually, they develop a bond like a father - son relationship.
The relationship would start off rocky. Snape would be resistant to Harry's kindness as he has always seen Harry as James Potter's son. But Harry's persistent efforts to understand and help Snape would gradually break down Snape's walls.