One common theme is sacrifice. Sirius is often willing to put his own life on the line to save Harry. For example, in many fics, he battles powerful dark forces without hesitation to protect the baby.
Another good one is 'Saving Baby Harry: Sirius's Mission'. In this story, Sirius defies all odds to reach Harry. He battles Death Eaters on his way to the Potter's place. Once he has Harry, he flees to a secret location known only to a few. The fic delves deep into Sirius's emotions during this time, from his fear for Harry's life to his determination to keep the baby safe.
Well, in some of these fics, Sirius might time - travel to save Harry from a future threat. For example, he could see a vision of Harry being killed by Voldemort in a new and terrifying way, and then he uses time - turning magic to go back and change the course of events.
Another theme is the power of love. Harry's love for Sirius gives him the strength and courage to face the many obstacles in his way. It's a theme that's often explored in fanfiction as it was also a big part of the original books. In these stories, Harry's love overcomes things like the dark magic surrounding Azkaban or the threats from Voldemort's followers.
One common theme is redemption. Sirius often feels guilty for not being there for Harry earlier, and by saving him from abuse, he is redeeming himself in a way. For example, in many stories, he reflects on how he could have prevented Harry from going to the Dursleys in the first place.
One great 'sirius saves harry fanfiction' is 'The Guardian's Promise'. In this fic, Sirius goes to extraordinary lengths to protect Harry from various dark threats at Hogwarts. He uses his skills as an Animagus and his knowledge of the wizarding world's secrets to keep Harry out of harm's way.
One great 'Harry Saves Sirius' fanfiction is 'The Rescue Mission' on FanFiction.net. It has a really intense plot where Harry uses his newly learned advanced magic skills to break into Azkaban and free Sirius. The author does a wonderful job of building up the suspense and the relationship between Harry and Sirius.
It's popular because it offers a different take on the relationship. Sirius is a complex and beloved character, and seeing him interact with a baby Harry is both cute and full of potential for interesting stories.
Well, at first, Sirius would be in a state of shock and panic. But then he'd have to step up. Their relationship would shift from one of equals (in a sense, as Harry was growing up) to a caretaker - cared - for dynamic. Sirius would have to learn about Harry's likes and dislikes as a baby. He'd sing lullabies to him, comfort him when he cries. This new experience would make their bond stronger in a very different way. Harry might not remember all of it when he's turned back, but there would be a subconscious change in how he feels towards Sirius, a deeper sense of security and love that stems from those baby - days when Sirius was his whole world.
Sirius could use his knowledge of the wizarding world's secrets. He might know some hidden places or spells that could keep Voldemort at bay. For example, he could put a powerful protective charm on the place where Harry is staying.