I think that the importance of second chances comes up often and is an important theme in this book. We see it in a couple of places and affects a couple of the major characters in this book. It affects Snape, Dumbledore, and even Draco Malfoy a little later in the book. Earlier in the series, in the Goblet of Fire, some of the characters speak of Dumbledore and his unusual belief in second chances.
One glaring example of second chances in this book is Snape. In his case, Dumbledore gave him a second chance as he believed that Snape, although a former Death Eater, changed. He became one of the heroes in this book which is interesting because most of us grew to dislike him throughout the books. Later in this book, we learn that he totally changed and centered his life around protecting Harry after Lord Voldemort killed Harry's mom Lily. This character change showed us a lot about how Dumbledore is right and that some people do deserve second chances. It also showed how wise Dumbledore is because Snape was invaluable to Dumbledore throughout the books as a spy against Lord Voldemort and the protector of Harry. It is interesting how JK Rowling painted Snape as a villain and was hated by Harry throughout the series.
Later in the book, there is a scene where Harry took what he learned from Dumbledore and put it into action. When Malfoy, Crabbe, Goyle, Hermione, Ron, and Harry were in the Room of Requirement together, Crabbe or Goyle released Fiendfyre (sorry, I don't remember which). When this happened, Hermione and Ron were able to escape, but Malfoy was not so lucky. Despite everything that Malfoy had done to Harry throughout the books, Harry gave him a second chance and saved him. I think that this really showed the type of character that Harry is and how Dumledore had an amazing influence on the decisions that Harry made. Would you have saved Malfoy? (NO WAY!!!!!!!!!!!!!!!!!!)
And finally we have Dumbledore. There is the book The Life and Lies of Albus Dumbledore written by Rita Skeetter that tests Harry's faith in Albus Dumbledore. A big question that is brought up at this point in the book is the question of if Harry will give Dumbledore a second chance and try to follow through on what he planned for him. Harry had to trust Dumbledore and the fact that he knew what he was doing and believe that Dumbledore was able to change from his youth. Would you have been able to if you were Harry?
(By the way have you guys noticed how quickly Harry seems to forgive people? Example: Ron)