Running a kite is symbolic of a state of innocence that can only be gained by serving others. At the beginning of the story, Hassan is the innocent character, and he serves Amir. His running of the kite represents this servitude. When the story concludes, Amir is running the kite for Hassan's son. He has cleansed himself of his past sins by saving Sohrab from the clutches of the psycophathic Assef. The central focus of his life has become fulfilling the needs of Sohrab.
The story is named "The Kite Runner" because throughout it, Amir is continually striving to force his cowardly instincts to obey his valiant goals. He wants to redeem himself of his past sins, and return to a state of purety and innocence where he gains satisfaction from serving others.