Well, Snape saving Harry who he sees as his abused son could start with him seeing the signs that others miss. Snape, being a complex character, might struggle with his own feelings at first. But then he'd use his skills and knowledge to help Harry. He could teach Harry some advanced magic to defend himself. And he might confront Dumbledore about the way Harry has been treated. Eventually, this could lead to a big change in the Hogwarts environment for Harry.
In some versions, Snape might notice Harry's withdrawn behavior or signs of physical harm during Potions class. Maybe Harry flinches when someone gets too close, which makes Snape suspect something is wrong.
Snape might be motivated by a desire for redemption. He has a complicated past, and saving Harry could be his way of making up for all the bad things he has done or been associated with. Additionally, he might see a bit of himself in Harry's situation, as he also had a difficult childhood, and this empathy drives him to act.
He may also get angry. Harry has a strong sense of justice and self - worth. Snape's rejection could be seen as unfair in his eyes, especially if he has started to develop some sort of positive feelings towards Snape. So, he might lash out at Snape or even try to prove him wrong by becoming more independent and successful on his own.
Well, it could become more positive. Harry might grow up seeing Snape as a hero or a protector, instead of the antagonist he was in the original story. This would lead to a more trusting relationship between them.
Harry could react with anger. He is a strong - willed character. So, he might confront Snape about his hatred. He would demand to know why Snape is treating him this way when they should have a father - son relationship. This could lead to a series of intense arguments between them.
The son could be a very different character from Harry. Instead of being the 'chosen one', he might be more of an outcast like Snape was. He could struggle with making friends at Hogwarts due to his father's reputation. However, he might also have a hidden talent for something unique, like ancient runes or arithmancy. And Snape, in his own way, would try to protect his son from the harsh judgments of the wizarding community.
Snape's reaction could be quite mixed. On one hand, he might be angry at Dumbledore for not doing more if he knew about the abuse. He may also feel a strange sort of empathy towards Harry. After all, Snape himself had a difficult childhood. So, he could start to be a bit more lenient with Harry in his classes and try to subtly offer support.
Well, it's mainly because it shows a different side of Snape. In the original series, Snape is often seen as a complex and somewhat antagonistic character towards Harry. In these fanfictions, his act of saving Harry gives him a chance at redemption. It also adds an interesting dynamic to their relationship, which was mostly strained in the books.
One popular one might be 'Snape's Redemption: Saving Harry'. In this fanfic, Snape discovers the extent of Harry's abuse early on and takes it upon himself to protect Harry from further harm. He uses his position at Hogwarts to keep a close eye on Harry and shields him from the Dursleys' wrath when they try to interfere with Harry's time at school.
Well, he could use his magic skills secretly. Maybe he casts a spell on the Dursleys to make them docile or distracted long enough to get Harry out. For example, he might use a Confundus Charm on them so they don't notice Harry leaving.