He could develop more patience. Harry, being a curious and sometimes headstrong boy, would test Snape's limits. But as he grows into the role of a father figure, Snape learns to be more patient and understanding, which is a big change from his usually short - tempered self in the original series.
A possible plotline is that Snape's adoption of Harry causes a rift in the wizarding community. Some might support it, seeing it as a way to keep Harry safe and under the watchful eye of a powerful wizard. Others might oppose it, fearing that Snape's dark past could influence Harry negatively. This conflict could lead to many adventures and challenges for both Snape and Harry as they try to navigate through the political and social turmoil.
In such a fanfiction, the plot could involve Snape taking on the role of a father figure to Harry and Draco. He might use his knowledge and skills to protect them from the various threats in the wizarding world. Snape could also teach them advanced magic, which they might not learn otherwise in Hogwarts. And there would likely be a lot of emotional growth for both Harry and Draco as they deal with their new family situation and their past traumas.
The concept is popular because it's a fresh take on the relationship between Snape and Harry. In the original series, their relationship is complex and often antagonistic. This idea of Snape adopting Harry flips that on its head and allows for a more positive and nurturing relationship to develop.
I'm not sure about a specific title, but generally in these types of fan fictions, the story often focuses on how Snape's adoption of Harry changes the course of Harry's life at Hogwarts. For example, Snape might use his knowledge of dark arts and potions to train Harry in a more private and intensive way. This could lead to Harry being better equipped to face Voldemort earlier. Also, the emotional aspect of their relationship, from Snape's initial reluctance to a growing father - like bond, is usually explored in great detail.
One common theme is redemption. Snape, who has had a rather dark past in the original series, gets a chance to redeem himself through his relationship with Harry. He can correct the mistakes he made in the past, especially regarding his treatment of Harry.
One popular fan fiction could be 'Snape's New Family'. In this fic, Snape's decision to adopt Harry leads to a whole new dynamic at Hogwarts. Harry experiences a different kind of mentorship from Snape, which is far from their usual tense relationship in the original series. It delves deep into how Snape tries to protect Harry while also teaching him advanced magic.
Snape becomes more empathetic. He starts to see Harry not just as James Potter's son but as an individual with his own struggles. For instance, he might notice how Harry is affected by the loss of his parents more deeply than he thought.
In this fan fiction, Snape could save Harry from Voldemort's followers who are after him. The blood adoption might be a way to give Harry a new family connection. Snape, having complex feelings for Harry's mother, might see this as a way to fulfill a promise or a way to atone for past actions. He could then teach Harry some of his more advanced potions knowledge and help him develop his occlumency skills better.
In this fan fiction, Snape's character development could be quite complex. Saving Harry is just the starting point. With the blood adoption, he is thrust into a new role. He may initially struggle with this new identity as a sort of surrogate father. However, as time passes, he could learn to love Harry in his own way. His development might also involve him reconciling his past actions, especially those that were harmful to Harry in the past. He could become a more well - rounded character, showing both his old prickly side when dealing with others at Hogwarts, but also a soft, caring side when it comes to Harry.