I'm convinced that the kite symbolizes redemption and dignity. So much of the book deals with "becoming good again" or overcoming shame and reviving your spirit. At the closing scene in the book, Amir and Sohrab fly a kite, but on a different level, it is an activity that allows them to move forward in life, past the pain and shame of events they've experienced. For Sohrab, he must heal from the pain of sexual abuse, the death of his parents, adjust to life in America, and allow Amir and Soroya to be close to him. For Amir, the kite symbolized his failed attempts to be loved by his father, but it also symbolizes his ability to stand up for himself and "become good again". By adopting Sohrab, he is able to atone for the cowardly act he committed toward Hassan when they were children living in Kabul.
The kite symbolizes Amir's happier moments in his life. The first, he is flying a kite with Hassan, and it is the last happy moment he will have with him.
The second time Amir is flying a kite, he is with Hassan's son, who he now raises, and is finally happy after a long and traumatic battle for the boy's life.