Why do you think Khaled Hosseini use the name The Kite Runner?
Kite running operates as a plot element and a powerful symbol of several themes of the story, themes of love, betrayal, culture, and hope.
First, as the previous responder noted, kite running is an activity that shows the bond between the boys. Kite running also represents Amir's triumph, a triumph that he offers his father, hoping for some sign of his love and appreciation. But this is a also a representation of Amir's failure because he does not help Hassan when Hassan is attacked. Kite running in this instance is a symbol of all that is good about Hassan and all that is cowardly about Amir.
Kite running is a symbol of Afghan culture as well. Notice that it is banned at various periods during the book for political reasons. But this is a portable attribute of Afghan culture, so Amir is able to partake in the tradition when he moves to America.
Kite running also acts as a symbol of redemption and hope. A kite is a symbol of our aspirations because it reaches for the sky. At the end of the story, what is happening when Sohrab begins to heal? It is kite running, which shows us there is hope for Sohrab, hope for Afghan culture, and hope for Amir in this new world.
I think he used the title "The Kite Runner" because it was the activity of running for and chasing kites that bonded the two main characters together. The two boys were socially different from each other, but were allowed to grow up as brothers and share in the same experiences.
More importantly, was this shared experience of running for kites together that ultimately tore the boys apart and altered their lives forever.