In one sense, Hassan is not overly courageous in The Kite Runner by Khaled Hosseini because everything he does is done out of love for the people who are important to him.
In another sense, Hassan is the most courageous character in the novel. He is noble, kindhearted, and selfless, so it is not surprising that he does things which are courageous.
First of all, on the day he was born his mother laughed at Hassan and left a few days later.
Sanaubar had taken one glance at the baby in Ali's arms, seen the cleft lip, and barked a bitter laughter. [...]. She had refused to even hold Hassan, and just five days later she was gone.
Hassan grows up motherless and with a harelip, two things which put him at a disadvantage, even for a Hazara. And, of course, he is a Hazara and a servant, two things which put him on the lowest rung of Afghanistan society.
Specifically, Hassan allows himself to be brutally assaulted by the bully Assef just so he can get the blue flag for Amir. Even more, he says nothing about it to Amir, not one word, though of course he knows that Amir could have helped and did not.
He is also courageous when he lies and takes the blame for stealing Amir's money and watch. He should have stood up for himself, of course, but he is willing to be sent away (with his father) to spare Amir any more pain (which Amir caused for himself).
When Hassan's mother, Sanaubar, appears after being absent for decades, Hassan accepts her. It does take him just a bit to conquer his feelings, but he is gracious and kind to her until the day he dies.
Hassan is courageous when he comes back to Kabul at Rahim Khan's request. He could easily have stayed in his small village and escaped the Taliban trouble in the city; however, he comes to try to help save Baba and Amir's house. It is an act that gets him killed.
Hassan would probably not call anything he does courageous; however, it is clear that he lived a consistent life of courage.