It's kind of a no-brainer, but Khaled Hosseini's follow-up to The Kite Runner, A Thousand Splendid Suns, also explores themes of friendship and betrayal. It, too, is set in Afghanistan, but the main characters are women. If you are a fan of TKR, I highly recommend ATSS.
For me, the theme of friendship and loyalty is a vital one from Hosseini's work. I think that I see this theme in a couple of other works and contrasting the views presented on this theme could be interesting. One of these works was the subject of a question some time back. Steinbeck's Of Mice and Men is a great conception of friendship to contrast with The Kite Runner. How does friendship look in both? What does it mean to stand up for your friend? How does loyalty look in both works? I think that this is one area of interest because it looks so different in both. Amir goes back to "become good again," while George will never be able to do this at the end of the work. I think that the examination of which friendship represents more qualities of a "good friendship" is something that can brought out from both works.
Another work that immediately reminds me of the concept of friendship is the King short story, "Rita Hayworth and the Shawshank Redemption." Here again, I think that the definition of what makes a good friendship is evident. I like how King was able to construct the reality of how the worst of conditions can forge the best of friendships. The presence of contingency does not take away from universality. This is something very important that he conveys and something that makes very good points of contrast with Amir and Hassan and George and Lennie. In all three of these works, the development of thematic concept of friendship is something that emerges.
I am studying 'In the Country of Men' and is very similar (in setting, plot and theme) to 'The Kite Runner'.
You could try Deborah Ellis' novels.. like Parvana.. it's about a girl's survival in Afghanistan and you can link both texts in terms of war, violence, responsibility, redemption, among others. Rita Hayworth and the Shawshank Redemption is also a good book to compare with.