One plot twist could be that Snape discovers a hidden secret about Harry's family that changes his whole perception of them. For example, he might find out that there was a prophecy related to Harry that was different from what everyone thought.
A possible plot twist is that Snape is actually being set up when he's sent to check on the Harry family. Someone might be using him to get to Harry or to cause trouble within the wizarding world. So, Snape has to figure out who is behind this while dealing with the Harry family.
In this fanfiction, a plot twist could be that Snape's feelings for Lily Potter resurface in a new way as he checks on her son Harry and his family. This could lead to him making decisions that go against what he was originally sent to do, like protecting Harry at all costs instead of just observing, which would completely change the direction of the story.
In the 'snape is sent to check on harry family' fanfiction, Snape's relationship with the Harry family could be full of twists. At first, he might be there just to do his job. However, as he gets closer to them, he could find himself protecting them in his own way. He might even become a sort of mentor figure to Harry, teaching him things that Dumbledore or others haven't, because he has a unique perspective from being sent to check on them.
It could start with a lot of mistrust from Harry's family towards Snape. But as Snape shows his true intentions (if he has good ones), they might gradually warm up to him. For example, Harry's mother could see some of the good in Snape that she knew from their childhood days.
One possible plot twist could be that Snape's secret family has a hidden magical ability that is crucial to the wizarding world. Maybe his family has the power to reverse dark magic spells, and this is something Voldemort has been searching for all along.
A possible plot twist could be that Snape is actually the one who arranged for Harry to be sent to Azkaban, but for a noble reason. Maybe he knew that Harry was in danger from a more powerful dark force and Azkaban was the only place that could keep him safe temporarily. Another twist could be that the real culprit behind the crime Harry is accused of is someone very close to Snape, and Snape has to choose between protecting that person and helping Harry. It could also be that Snape discovers that Harry has a hidden power while in Azkaban that could change the fate of the wizarding world, and he now has to keep Harry alive at all costs.
One plot twist could be that Snape is actually mentoring Harry on behalf of Dumbledore from beyond the grave. Maybe Dumbledore had some secret plan that he didn't get to fully execute, and Snape is now following through with it. Another could be that Snape discovers a new magical ability in Harry that no one else knew about.
Well, Snape could be initially very guarded. He would be cautious about approaching Harry's family. Since he has his own secrets and his loyalty to Dumbledore might be in conflict with his personal feelings towards the Potters. He might approach the task with a sense of duty but also a great deal of wariness.
The plot could take a really interesting turn when it comes to Voldemort. Since Snape is raising Harry, he might train Harry in a way that Voldemort doesn't expect. Maybe Harry would be more of a spy in the dark side's camp. Snape could also use his position to mislead Voldemort about Harry's true abilities and intentions. This would create a whole new set of challenges and power dynamics in the fight against the Dark Lord. Harry's loyalties would be constantly tested, and his relationship with Snape would be even more crucial to the story.
One possible plot twist could be that Snape saves Harry in a very public way, which shocks everyone in Hogwarts. Maybe Umbridge was about to use an illegal and extremely harmful spell on Harry during a public 'interrogation' session, and Snape steps in at the last moment, revealing some long - hidden information about Umbridge's true nature that even Dumbledore wasn't fully aware of.
One plot twist could be that Snape discovers Harry is his son right before a major battle. This new knowledge gives him the extra motivation to fight even harder to protect Harry. Another could be that Voldemort finds out about this relationship and tries to use it against them both, perhaps by threatening to harm Harry if Snape doesn't turn on Dumbledore.
One possible plot twist could be that when Harry goes back, he discovers that Snape's situation is far more complex than he initially thought. Maybe Snape was involved in some secret mission that put him in danger, and Harry's interference actually endangers the entire wizarding world in unexpected ways. For example, Snape might be on the verge of uncovering a new dark magic plot that Harry's presence disrupts.