Power is a major theme. Harry being in a position of power as Lord Azkaban has to learn how to use it responsibly. Another theme could be redemption, as he might try to redeem the prisoners or himself in some way.
One popular 'Harry is Lord Azkaban' fanfiction might be 'The Ascent of Harry in Azkaban'. It has an interesting take on how Harry gains power and influence within the prison. The story is full of mystery and unexpected plot twists.
Dumbledore might play an important role as well. He would be guiding Harry through this new and strange situation. He could have some secrets about the Lord of Azkaban that he reveals to Harry over time. Also, some of the Azkaban guards or prisoners could become significant characters as Harry tries to reshape Azkaban.
Harry Potter himself is of course very common. He's usually the central figure in these fanfictions. Hermione Granger is also often included as she is Harry's friend and a brilliant witch. Ron Weasley too, as he's part of the trio. Sirius Black might also be a common character since he has a connection to Azkaban.
One interesting plot could be a new adventure where Harry discovers a hidden chamber in Azkaban that holds a powerful magical artifact. This artifact might be related to an ancient wizarding family and its secrets. Another plot could involve the Dementors having a different origin than what is known in the original series, and Harry has to find a way to stop them from spreading a new kind of dark magic.
Harry might survive in Azkaban because of his connection to his friends, even if they betrayed him. Their memories could give him the strength to hold on. And he may be able to use his magic in unique ways. He could draw on the positive emotions he has left, like his love for his parents, to create a shield against the Dementors. Plus, he may be constantly looking for a way to prove his innocence and that hope could keep him alive.
In this fanfiction, perhaps a major plot point is Harry using his new position to reform Azkaban. He could introduce new ways of dealing with prisoners that are more humane. Also, his power might attract the attention of dark forces who want to either use him or destroy him, which would add a lot of action and suspense to the story.
He might use the invisibility cloak. Harry could sneak into Azkaban unnoticed and then use magic to break Draco out of his cell. The cloak has served him well in the past, and it would be a simple yet effective way to get past the guards.
It could be Harry Potter himself, of course. He might be on a mission to either free someone wrongly imprisoned in Caer Azkaban or to stop a new dark threat emerging from there. Hermione could also be a main character, using her intelligence to figure out the mysteries of Caer Azkaban.
Maybe he uses some powerful magic spells that he has learned secretly. For example, the Patronus charm could be used to fend off the Dementors which are the main guards in Azkaban. He might also get some unexpected help from his friends or some mysterious characters within the fanfiction world.