There is also a class and ethic dynamic in The Kite Runner that I believe is reflected in the use of this statement, first by Hassan to Amir and then later by Amir to Sohrab. Amir and Hassan are not from the same class or ethnic group, and this difference is central to the plot and themes.
Amir is a Pashtun, and Hassan is a Hazara. The Pashtuns are the ruling class in Afghanistan, and it is clear that the Hazaras are a lower class and ethic group, historically treated quite poorly, and also shown as treated quite poorly in the setting of the novel. Hassan and his father Ali are servants in the household of Baba and Amir, and while Amir and Hassan are raised together, there is a clear line in Amir's mind of their differences. Amir notes,
But in none of his stories did Baba ever refer to Ali as his friend. ... I never thought of Hassan and me as friends either (25).
Amir looks down upon Hassan, and the relationship is such that Hassan saying "For you, a thousand times over" is really what Amir expects from Hassan, because he perceives him to be an inferior servant. He really does not see that Hassan is a true friend and is saying this out of love, not out of duty as a servant.
On his long journey of learning and repentance, as Amir finds himself and his father looked down upon in the new land, the shoe is on the other foot. They are subject to small, daily humiliations in the United States, for example, Baba's humiliation at collecting any kind of benefits and Amir's concern about not being quite good enough to wed Soraya, whose father is a general. Amir begins to understand what it feels like to be regarded as inferior. By the time he rescues Sohrab and says to him, "For you, a thousand times over," this is meant to represent not only his repentance for what he had done to Sohrab's father, but also his understanding that we must all be servants to one another and that class and ethnicity should play no part in our willingness to do so. He is saying this out of love and friendship, as Hassan had said it to him.
So, while there is no question that the use of this declaration represents Amir's repentance for all he has done wrong to Hassan (and Ali), I do think that the class and ethnic tension are part of the meaning behind its use, too.