Another example of redemption in Harry Potter and the Prisoner of Azkaban is Buckbeak the hippogriff. Buckbeak is really an elegant creature that requires certain rules be followed when approaching him. Malfoy does not believe Hagrid when warned to stay away from him so he runs up to Buckbeak and in turn gets thrown to the ground and is mildly injured. Of course Malfoy wants the creature destroyed so he complains to his father and Buckbeak is sentenced to death. Harry, Ron, Hermione, Hagrid and Dumbledore all know that Buckbeak is innocent and in the end free him and he carries Sirius off to freedom.
I guess I would pick Sirius Black and Remus Lupin as the examples of redemption in this book.
I think that you can argue that Lupin is redeemed to some extent in this book. He has been shunned because he is a werewolf but Dumbledore brings him back to teach. Of course, he loses his job at the end of the year, but in later books he'll be back and so he's clearly rejoining society to some degree.
More than that, we see the redemption of Sirius Black, at least in the eyes of Harry, Ron and Hermione. Instead of being a hated and feared Deatheater, he is now Harry's beloved godfather.
I suppose you can say that Crookshanks is redeemed over the course of the book as well. At first he seems like the real bad guy always trying to kill Scabbers. But then at the end he is revealed as a hero of sorts because he realized what Scabbers really was.
I disagree redemption is a theme in the story because the evidence for it is too weak. The strongest example is of Sirius's case, but (1) he was innocent of wrongdoing that needs forgiveness, and (2) few people finally understand his innocence (while almost everyone continues to wrongly view him as deserving a Dementor's kiss). Similarly, Buckbeak and Crookshanks do no wrong that they need redemption for. As Hermione points out about her cat, he "doesn't understand it's wrong!" Animals act on instinct, so it makes no sense to apply human moral judgments to their actions. The same is true of Buckbeak's attack on Malfoy when he is provoked by that git's insult. As Hagrid understands, it is the nature of Hippogriffs to be proud and respond viciously to an offense; they can't help it. The examples of the animals in the story help illustrate what's different about cruel behavior by humans: we have a concept of morality and can choose how we act. With Buckbeak, we can say revenge is natural and not question the justice of clawing Malfoy to avenge mere insulting words. For humans, it not so simple.
Another problem with the idea of redemption as a theme is that it understands the term very narrowly. Sirius and Buckbeak are simply rescued from others' cruelty and injustice, so it would be more precise to say simply that the theme of the story is "being rescued," which sounds ridiculous. "Redemption" can also mean (1) atonement for guilt or (2) deliverance from sin; but neither of these meanings make sense in relation to Sirius, Buckbeak, or Crookshanks.
Furthermore, it is always vague and reductive to speak of a theme using just one word. A theme is an important idea suggested in a story, not a general subject. People often confuse "theme" and "topic," perhaps partly because it is much easier to speak vaguely than it is to be precise.
With regard to this story, the main theme certainly has to do with revenge. The theme could be stated in different ways, but it might be expressed as the idea that revenge is cruel and unjust. How many characters in this story have "got it in for" somebody? In addition to his usual grudge against Harry, Snape has it in for Neville, Lupin, and Sirius--even to the extent of wanting to kill Sirius to avenge a stupid prank played in school. Buckbeak takes revenge on Malfoy. Malfoy takes revenge in turn by trying to get Hagrid sacked and Buckbeak killed. The Ministry wants to punish Sirius with a Dementor's kiss, an extremely evil sort of weapon that Lupin suggests nobody could really deserve. When Harry learns that his parents were betrayed to their deaths by Black, even he broods on revenge and thinks he deserves the Dementor's kiss. Yet, we learn that they are all wrong in judging Sirius, and also that it can be extremely difficult to convince people of the truth. As Dumbledore says, even he has no power to do this. The story shows how ignorant people are and prone to mistakes of judgment, and that those mistakes are often prompted by personal motives such as pride, fear, and hatred.
In the end, Harry stops Lupin and Sirius from killing Pettigrew even though the latter is an utterly despicable person who is unquestionably guilty of murdering numerous people. Harry doesn't forgive Pettigrew. As he says, he doesn't want his father's best friends to be killers. Revenge is not justice.
Sirius Black's story would definitely be the main example of redemption in this book. He was feared as the most notorious criminal in the Wizarding World, and was even blamed for the murders of his closest friends, James and Lily Potter. The truth is finally revealed to Harry and his friends, as well as Dumbledore, and the readers discover that Sirius is in fact innocent and very willing to take care of his Harry, who is in fact his godson.
The second example of redemption I can think of Crookshanks. Most of the characters believe that Crookshanks pursued Ron's rat Scabbers for no particular reason and that Crookshanks was responsible for Scabbers' apparent death. It is finally revealed, however, that Croonkshanks was innocent and that Scabbers is actually the Death Eater who had betrayed Harry's parents to Voldemort.