In some fanfictions, a plot twist could be that the Ministry of Magic has been secretly working with Voldemort's followers to keep Sirius in Azkaban. And Harry, Ron, and Hermione find out about this conspiracy. They then have to not only break Sirius out but also expose the Ministry's dark secrets. This would add a whole new layer of political intrigue and danger to the story.
A cool plot twist might be that the time - turner Hermione uses has some unforeseen side effects. Instead of just allowing her to attend more classes, it starts to create glitches in time that cause different events from the past to bleed into the present. For instance, old spells from the founders of Hogwarts start to randomly appear in the corridors, and the characters have to figure out how to stop this chaos while also dealing with the Sirius Black situation.
Obviously, Harry Potter is one of the main characters. He is likely to be the driving force behind the attempt to get Sirius Black out of Azkaban.
One interesting plot twist could be that Sirius Black is actually being framed by a more powerful dark wizard who has the ability to disguise as others. This would add an extra layer of mystery.
In some fanfics, Lupin might turn out to be under the influence of a spell that makes him seem more dangerous than he actually is. Maybe the real threat is a hidden character who is using Lupin's condition to create chaos. For example, they could be making him lose control of his werewolf form more often than normal, which would lead Harry and his friends to suspect Lupin wrongly.
One major plot twist is that Sirius Black is not the villain everyone thinks he is. He was wrongly accused of betraying Harry's parents. Another is that Peter Pettigrew, who was thought to be dead, is actually alive and was the real traitor.
One possible plot twist could be that Neville discovers a new and powerful spell that turns the tide of the battle. Another could be that Draco Malfoy has a change of heart and secretly helps the Order of the Phoenix during the final battle.
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.
In some fanfics, he might use powerful magic that he has secretly mastered. Maybe a new spell that can create a portal right out of his cell.
One possible plotline could be Harry discovers a secret passage that leads out of Hogwarts in a completely unexpected place, like inside the Room of Requirement. Another might be that he has to use a new spell he learned from an old book to break through the magical barriers around Hogwarts while escaping.
One possibility is that Dumbledore discovers new evidence that clears Harry's name. Maybe there was a misused spell by the Ministry that made it seem like Harry was guilty. Dumbledore then uses his influence and magic to break Harry out of Azkaban and they begin to expose the real culprits behind the false accusations.