One interesting plot could be that Harry uses his knowledge of the future to help Snape avoid some of the bullying he endured. For example, he might warn Snape about certain Slytherins who would target him. Another could be that they work together to change some small events that have a big impact on the overall wizarding world's future. Harry might also introduce Snape to some of his future friends in a different context, like showing Hermione's intelligence to Snape much earlier.
It would definitely shake things up. The Gryffindors and Slytherins have a long - standing rivalry. With Harry, a well - known Gryffindor, befriending Snape, a Slytherin, it could start to break down some of those barriers. Students might be more open to interacting with those from different houses.
He might use some powerful magic artifacts. For example, the Time - Turner, if he can get his hands on an even more powerful or modified version of it. This could allow him to bypass some of the normal time - travel restrictions.
One possible plot twist could be that when Harry goes back, he discovers that Snape's situation is far more complex than he initially thought. Maybe Snape was involved in some secret mission that put him in danger, and Harry's interference actually endangers the entire wizarding world in unexpected ways. For example, Snape might be on the verge of uncovering a new dark magic plot that Harry's presence disrupts.
One possible plot twist could be that Snape, in his attempt to save Harry, accidentally changes the entire future of the wizarding world. For example, he might prevent Voldemort from ever rising to power in the first place, which would mean that all the relationships and events that we know from the original story would be completely different. Another plot twist could be that Snape discovers that saving Harry is not as straightforward as he thought. Maybe there are dark forces at work that are trying to stop him from changing the past, and he has to fight against them while also trying to protect Harry.
Snape's character development could take a very different turn. He may start to value life more, not just Harry's but also those around him. In the original, his focus was mainly on his own pain and his loyalty to Dumbledore. But going back in time, he realizes that every action he takes can have a huge impact on the future. This could lead to him becoming a more empathetic character. He might also become more of a hero in the eyes of others, which is a big change from his somewhat villainous reputation in the original story. He could start to see himself in a new light and strive to be a better person not just for Harry's sake but for the sake of the entire wizarding world.
The Hogwarts setting also plays an important role. The familiar corridors, classrooms, and the overall atmosphere of the school provide the backdrop for their interactions. It could be a chance encounter in the Potions classroom or a meeting in the Forbidden Forest. These settings add to the authenticity of their budding friendship as it's within the world they both know and are a part of.
One interesting plot could be Snape going back and trying to prevent the death of Lily. He might use his knowledge of the future to change certain events that led to her demise. Maybe he would try to expose Pettigrew earlier or find a way to make James Potter more cautious.
One popular 'Harry befriends Snape fanfiction' is 'The Unlikely Bond'. It delves into how Harry, after seeing a different side of Snape during a difficult time at Hogwarts, decides to reach out. Their friendship starts tentatively as they discover they have more in common than they thought, like their love for potions in different ways.
Bellatrix Lestrange could also be a character who interacts with Snape. If Snape goes back far enough, he might try to influence Bellatrix's path and keep her from becoming such a devoted Death Eater. This could have a major impact on Voldemort's power and the overall balance of the wizarding world.