Harry could also show his rebellious side similar to Sirius by befriending those who are considered outcasts. Sirius was friends with werewolves and other marginalized magical beings. So Harry might befriend students from different houses who are not well - liked, or those with a bit of a dark family history, just like Sirius did with his group of friends back in the day.
He could be more defiant towards the authority figures he doesn't trust. Sirius was always a bit of a rebel against the Ministry when they were wrong. So Harry might openly question the decisions of some of the Hogwarts teachers or the Ministry officials who visit the school, if he feels they are being unjust, much like Sirius would have.
Well, Harry might start making quick - witted remarks just like Sirius. Sirius was known for his sharp and often sarcastic humor. So Harry could be cracking jokes at the expense of the Slytherins or making light - hearted fun of Hogwarts' rules, similar to how Sirius used to do when he was a student.
One interesting storyline could be Harry taking on Sirius' sense of adventure. He might start exploring hidden parts of Hogwarts, like Sirius did when he was a student. For example, he could search for secret passages that Sirius might have known about but never fully revealed. Another could be Harry adopting Sirius' more rebellious attitude towards the Ministry. He could publicly oppose some of their unjust rules and regulations, much like Sirius was always a bit of an outlier in the wizarding world.
His relationship with Dumbledore could become more complex. Dumbledore knew Sirius well, and if Harry starts emulating Sirius' more impulsive side, Dumbledore might have to be more hands - on in guiding Harry. On the other hand, among the other Gryffindors, Harry could become a sort of leader figure. Since Sirius was a popular and respected Gryffindor in his time, Harry acting like him could inspire his housemates and lead to more group activities and a stronger sense of unity within Gryffindor.
Sirius could help Harry cope by sharing his own experiences of being an outcast in his family. He could tell Harry that just because his parents did this horrible thing, it doesn't mean he's unlovable. Sirius could also introduce Harry to his friends in the wizarding world, like Remus, who could also offer support and friendship.
It's possible that in the cave, they start to see each other as equals rather than just a mentor - student relationship. They might argue about the best way to deal with the cave's mysteries at first, but through that process, they learn to value each other's perspectives. Also, if one of them gets injured in the cave, the care and concern shown by the other would enhance their relationship.
There are quite a few. One such fanfic is 'The New Guardians' where Sirius takes Harry under his wing to protect him from new threats.
One reason could be Harry's impulsiveness. Harry often jumps into dangerous situations without thinking, and Sirius, being his godfather and having a protective nature, would shout at him to make him realize the risks. For instance, if Harry decides to go after a Death Eater alone.
It may explore the emotional bond between them. Sirius could be like a father figure to Harry. He gives Harry advice and guidance, which is different from what Harry gets from others. This special connection makes Harry trust Sirius more than anyone else.
Maybe Sirius gets help from some powerful wizards or witches who believe in his innocence. They might testify on his behalf in front of the wizarding authorities. Sirius also could show his strong emotional bond with Harry. He has always been there for Harry in spirit, and now he can be there physically to raise him, which convinces the authorities to give him custody.
Sirius helps Harry become more confident. By sharing his knowledge and experiences, Harry feels more capable. For example, when Sirius teaches Harry a difficult spell, Harry realizes he can master things he thought were out of his reach.