Snape might show a softer side earlier in the story. Usually, he's so guarded, but at Christmas, he could let his defenses down a bit. Maybe he shares a memory of his own Christmas with Lily when they were children. This would give the reader a different perspective on his character.
He could be developed as more of a mentor figure during Christmas. In the normal Hogwarts setting, he's often seen as an antagonist to Harry. But in this fanfic, he could take Harry under his wing during the Christmas break. He might teach Harry some advanced potions techniques or share some wisdom about the wizarding world that he wouldn't usually disclose. This would show that there's more to Snape than just his prickly exterior.
Snape would have more opportunities to right his wrongs. For example, he could go back and be more honest with Lily about his feelings. His character could become less bitter as he gets a chance to change things that made him so resentful in the first place.
In fanfic, their relationship can be less hostile. For example, Snape may not be as harsh with Harry in class as he was in the books. He could be more understanding of Harry's situation, especially considering Harry's role in the fight against Voldemort.
Snape's character might develop more positively. Without the burden of his past mistakes and betrayals haunting him, he could be more confident. He might be more open to forming friendships, especially with Dumbledore. Their relationship could start on a more equal footing, leading to a more trusting bond.
Harry might become more introverted and studious. Since Snape is known for his academic prowess, Harry could inherit some of that trait. He might also be more distrustful of others at first, given Snape's own guarded nature.
The relationship can be developed by having Snape protect Harry in unexpected ways. Maybe in a fanfic, there's a plot where some dark wizards are after Harry. Snape, secretly, uses his skills to keep Harry safe. Later, when Harry discovers this, Snape has to explain his actions. This explanation can be a turning point where they start to trust each other more. Snape can then start to offer words of comfort based on his understanding of Harry's situation. They could also start to work together on some small tasks, like researching a new potion that might help Harry in his fight against the dark side, which further cements their relationship.
Well, in these fanfictions, Snape often becomes more openly affectionate towards Harry. Instead of hiding his feelings behind a mask of coldness, he gradually lets his guard down. For example, he might start giving Harry small gifts like a special quill or a book on advanced magic that was once important to him.
Some fanfics make Snape more openly vulnerable. In the original books, he's very guarded, but in fanfics, he might show his true feelings more easily. For example, he could be more open about his love for Lily to other characters.
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 some fanfictions, Snape's relationship with Harry is shown as more of a father - son relationship. For example, in 'A Different Kind of Bond', Snape takes Harry under his wing early on, and they develop a mutual respect. This is very different from the strained relationship in the books.
Snape's character often develops from a cold and aloof figure to a more caring and father - like one. He starts to see Harry not just as James Potter's son but as an individual he is responsible for. His protective instincts grow, and he becomes more patient. For example, he might initially snap at Harry for small mistakes but over time, he learns to guide him gently.