He could be constantly thinking about his godson, Harry. The thought of seeing Harry again and protecting him from Voldemort would be a huge motivation. Sirius would probably replay in his mind all the things he wants to teach Harry and all the ways he can be there for him. This would keep his mind focused on the future outside Azkaban and give him the determination to escape.
In some fanfics, Harry Potter might escape from Azkaban through powerful magic spells that he manages to conjure secretly. Maybe he uses a spell to create a diversion among the Dementors and then slips away while they're distracted.
One idea could be that Sirius has an unexpected ally inside Azkaban. Maybe a Dementor that has some remnant of humanity left and helps him in a way. Another could be that Sirius discovers a secret passage within Azkaban that was long forgotten. It could lead to a place outside the prison's normal boundaries.
In some fanfictions, Harry might use powerful magic he's learned secretly. For example, he could discover a new spell that can break the Azkaban's defenses. He then sneaks into Azkaban, locates Sirius, and uses this spell to free him from his cell. Maybe he also has to fight off some Dementors on the way out.
Yes, it's possible. Sirius is a popular character in the Harry Potter series, and many fans like to imagine him in different scenarios in fanfiction. He has a complex personality and a lot of untold stories, so fanfiction about him in the context of 'Prisoner of Azkaban' can be very interesting.
One could also write a fanfiction about Sirius Black's impact on Hogwarts during the events of Prisoner of Azkaban. For instance, his presence, even though he's on the run, could have a big influence on the students. Some students might start to question the Ministry of Magic's actions, and Sirius could inspire them to fight for justice.
In many fanfictions about Sirius Black in Prisoner of Azkaban, he is often given a chance for a new adventure. Writers might imagine him escaping Azkaban in more elaborate ways, or having different interactions with Harry Potter and his friends. For example, some fanfictions have him forming a secret alliance with other prisoners to break out together.
One great fanfiction could be 'Sirius's Redemption'. In this fic, Sirius, after escaping from Azkaban, tries to reintegrate into the wizarding world while dealing with his past traumas. It delves deep into his relationships with Harry and his old friends. The author does an amazing job of portraying Sirius's struggle with his sanity after years in that dark prison. Another interesting one is 'Sirius: A New Beginning'. It focuses on Sirius starting a new life outside of the shadow of Azkaban, exploring new magic and forming new alliances. It's a really engaging read if you like seeing the character grow in different directions.
One idea could be that Sirius Black's daughter, who has been hidden away all this time, discovers her true identity while trying to break into Azkaban to clear her father's name. She could have some unique magical abilities that help her in this dangerous quest.