One possible plot could be that Sirius manages to escape with Harry undetected. He could then raise Harry in secret, perhaps in a wizarding community on the outskirts. Sirius would teach Harry all about his parents and the wizarding world, but also keep him hidden from Voldemort's remaining followers. Harry would grow up with a strong sense of family and justice, having Sirius as his sole parental figure.
A great plot twist could be the appearance of unexpected allies or enemies. Maybe Lupin, who is also an outcast in some ways, shows up and has a different view on how Sirius should raise Harry. On the enemy side, Voldemort's followers might figure out that Sirius has Harry and target them both. Also, Sirius could find out that there are some special powers within Harry that are being suppressed by the wizarding world's norms, and he has to fight to let Harry develop those powers.
At first, it might be a bit rocky as Sirius adjusts to taking care of a baby. But as Harry grows, they would form a deep bond. Sirius would protect Harry at all costs, and Harry would come to understand Sirius's sacrifices for him. They might have some arguments as Harry gets older, especially when Harry wants to explore more of the wizarding world on his own, but their love for each other would always prevail. Sirius would be the one Harry turns to for advice, whether it's about girls or dealing with dark wizards.
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.
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.
A third one is 'Sirius's Adventure with Baby Harry'. In this story, Sirius takes baby Harry on little adventures around the wizarding world. They visit different magical places and meet interesting characters. It's a story that emphasizes their relationship and how it grows over time.
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.
Sirius would probably be very protective. He'd use his magic to create a safe and comfortable environment for Harry. For example, he might conjure up soft, warm bedding for the baby. He'd also try to find the best baby - wizarding supplies, like magic - infused milk bottles that ensure Harry gets all the nutrition he needs.
Sirius influences Harry's early life by being a protector. He stands as a shield against any potential threats. He might use his Patronus, a large dog, to guard Harry at night, making Harry feel safe in his sleep.