Well, it's mainly because it gives a new perspective on the characters. Snape was always a complex character in the original series, and this scenario allows us to see a softer, more caring side of him. For Harry, it's a chance for him to get the love and care he was denied at the Dursleys.
One popular one could be 'Snape's New Beginning'. In this fic, Snape discovers Harry's abuse and takes him under his wing. It shows a different side of Snape, more nurturing than in the original series. Another might be 'Snape and the Abused Boy'. It delves deep into how Snape helps Harry overcome his trauma while also dealing with his own inner demons.
There's also 'Snape's Redemption through Harry'. It delves deep into how Snape's decision to adopt Harry changes both of their lives. Snape has to deal with his own prejudices while protecting Harry from his abusers. It's a really interesting take on their relationship.
They're popular because it offers a different perspective on the Harry Potter universe. We get to see Snape in a more nurturing role which is quite different from his canon portrayal.
One common theme is redemption. Severus, who has had a complex and often negative relationship with Harry in the original series, gets a chance to redeem himself by protecting and caring for Harry. It shows that he can be more than just the strict and often cruel teacher.
You can start by looking on fanfiction.net. It has a large collection of Harry Potter fan fictions, and there are likely quite a few 'Snape adopts Harry' ones in there. Just use the search function and type in the relevant keywords.
One popular one is 'A Father's Love'. In this fanfic, Severus discovers Harry's abuse and takes him under his wing. He teaches Harry potions and helps him deal with the trauma of his past in a very heart - warming way.
In this fan fiction, Snape's character development is quite remarkable. At first, he might be hesitant and conflicted about saving and adopting Harry. But as time goes on, he becomes more selfless. He has to learn to communicate better with Harry, which is a big change for him. He also has to face his own demons from the past, like his relationship with James Potter. This journey of self - discovery and growth makes his character more three - dimensional and likable in the context of the fan - made story.
One big challenge is their pre - existing relationship. They've had a lot of conflicts in school. Snape has been rather harsh on Harry, so they need to overcome that animosity. Also, Snape has to deal with Harry's trauma from the Dursleys. He needs to help Harry heal and grow in a positive way, which is not easy for someone as emotionally reserved as Snape.
In this fan fiction, Snape might adopt Harry because he realizes that Harry has a lot of potential that is being stifled at the Dursleys. He could take Harry to a secret location and train him not only in magic but also in self - defense and survival skills. Snape would have to deal with his own complex feelings towards Harry, as he has both resentment and a sense of obligation. And Harry would have to adjust to living with Snape, who is not exactly the warmest person. It could be a story of growth and understanding between the two.