One motivation could be Snape's sense of duty as a teacher. Despite his dislike for Harry in some ways, he still has an obligation to protect the students. So, when he sees Lockhart endangering Harry, he steps in.
Another possibility is that Snape uses his position as a teacher. He might threaten Lockhart with the consequences of harming a student. Given Snape's intimidating presence and his own power within the school, Lockhart would be afraid to cross him. Snape could also call for other teachers to help, making it clear that he won't let Lockhart get away with any harm to Harry.
Love for Lily. Since Harry is Lily's son, Snape might feel an obligation to protect him. He couldn't save Lily, but he can try to keep her son safe.
Well, it could be a story exploring different relationships. Maybe it's about how Harry's interactions with Snape and Lockhart, especially in terms of touch, could signify different things. For example, a touch from Snape might be full of hidden meanings and complex emotions due to their strained relationship in the original series. And Lockhart's touch could be more of a self - centered or showy gesture. It could also be a fanfic that delves into a completely new adventure where touch plays a crucial role in magic spells or character development.
Another theme is the clash of personalities. Lockhart is flamboyant and self - centered while Snape is brooding and often secretive. Fanfics often play on this contrast, creating interesting situations and dialogues between them.
It could be that Snape wants to spite Vernon. Vernon represents everything that Snape despises about Muggles who are cruel and unaccepting. By saving Harry from Vernon, Snape is showing his superiority over Vernon and also his power as a wizard. And of course, there's still that underlying reason of his love for Lily which makes him want to protect her son.
In this type of fanfiction, Snape could simply intimidate Vernon. Snape has an air of mystery and a certain power about him. He might show up at Vernon's doorstep, give Vernon a cold stare, and with his reputation as a powerful wizard, Vernon might be too scared to oppose him. Snape could then just take Harry without much resistance from Vernon.
One possible plotline could be that Lockhart attempts to use Harry for some self - promoting stunt that goes horribly wrong, and Severus steps in to rescue Harry from the immediate danger. For example, Lockhart might try to make Harry participate in a dangerous magic display for his own glory, but Severus, seeing the real danger, intervenes.
One way is to make touch a form of magical communication. Harry could touch Snape's hand and suddenly be able to understand Snape's thoughts about protecting him all along. And when he touches Lockhart, he realizes Lockhart's false bravado. Another way is to have touch be a trigger for magical events. Maybe when Harry touches both Snape and Lockhart at the same time, it opens a portal to a new magical realm.
Well, perhaps Snape could use his authority as a teacher. He might see Ron being too pushy or possessive towards Harry in a romantic sense and intervene by giving Ron extra assignments or detentions to keep him away from Harry.
The Dursleys might be left in a state of shock and confusion. Since they have been so mean to Harry, they could be worried that the wizarding world will come after them for revenge. But since they are Muggles, the wizarding world may just leave them be, but they will always live in fear of the unknown.