Maybe Snape has some complex emotions while delivering the letter. He might be secretly hoping Harry won't come to Hogwarts because of his dislike for James Potter, yet he has to follow Dumbledore's instructions.
It could be Voldemort. Even though he doesn't directly appear in the act of letter - delivering, his influence might be hovering over Snape. Snape might be worried that by delivering the letter, he's bringing another potential threat to Voldemort, and Voldemort's shadowy presence could create conflict for Snape.
Well, in this fanfiction, Snape might be forced by some magical circumstances to deliver Harry's letter. Maybe Dumbledore has a secret plan and Snape is reluctantly following it. It could lead to some interesting interactions between Snape and Harry, perhaps Snape finally shows a bit of kindness that Harry never expected from him.
Dumbledore could be involved. He often has his fingers in many pies at Hogwarts, so he might be the one who arranges for Snape to deliver the letter to Harry for some greater good or part of his grand plan. Another possible character could be Hermione. She might notice something strange about Snape delivering the letter and start to investigate, suspecting that there's more to it than meets the eye.
The letter might be a summons to a secret wizarding council. Harry Snape's journey to deliver it could involve him facing his own inner demons as well as external obstacles. He might have to pass through dangerous terrains and encounter magical beings that are either trying to stop him or help him depending on their own agendas. As he gets closer to his destination, he discovers more about the true nature of the council and his role in the wizarding world.
Perhaps Snape has a moment of softness. He sees Harry as an innocent child about to embark on a great adventure. His own past experiences with Hogwarts make him want to be part of this important moment for Harry, despite his usual cold exterior towards the boy.
Well, it could potentially make their relationship less hostile from the start. Harry might see Snape in a more positive light, as Snape is doing something kind for him. Snape, on the other hand, might also feel a bit of a connection to Harry, knowing he's starting this important journey.
Well, it would change a great deal. For one, he might get better grades in Potions as Snape would be more likely to give him extra help. Also, he might be more protected from the likes of Draco Malfoy and his gang, since Snape would not tolerate any bullying towards his adopted son. Harry could also have access to Snape's private library, which would expand his knowledge of magic far beyond what he could learn in the regular Hogwarts curriculum.
Well, in fanfiction, if Snape raises Harry at Hogwarts, it would surely have an impact on his friendships. Ron, for example, might feel a bit left out at first. He was used to being Harry's closest friend, but with Snape taking on a more parental role, Harry might be spending more time with Snape, studying or learning things that Ron doesn't understand. Hermione, on the other hand, might be more understanding. She might encourage Harry to make the most of this situation. And some of Harry's other friends might be intimidated by Snape's presence in Harry's life. But overall, Harry's true friends would still be there for him, although the dynamics would be different.
Well, Harry might become a much more serious student. Snape, being strict, could push Harry to excel in Potions and Defence Against the Dark Arts even more. He might also instill in Harry a sense of discipline that he didn't have before. And Harry could gain a different perspective on Snape, perhaps seeing him as a father figure over time.