What is the significance of Hassan's son, Sohrab, in The Kite Runner?

When Amir and Hassan were children, they used to read and act out stories, and Hassan's most beloved was the 10th century Persian epic, the Shahnamah. Their favorite part of the book was that of "Rostam and Sohrab." It tells the story of the great warrior, Rostam, who slays his long-time opponent, Sohrab, in battle. As Sohrab lays dying, he reveals that he is actually Rostam's long-lost son.

The author symbolically weaves this story into the plot: Hassan represents Sohrab, and Amir is Rostam. Like Rostam and Sohrab, Hassan and Amir are unknowlingly related; and, in Amir's mind--and in his nightmares--it is Amir who causes the death of Hassan at the hands of the Taliban. Hassan names his son Sohrab because he loves the story (without knowing that he is in any way related to Amir), and author Khaled Hosseini uses it as an ironic and symbolic twist in The Kite Runner.

Who is Sohrab and what is his significance in The Kite Runner?

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Khaled Hosseini does not introduce Sohrab until late in The Kite Runner, so he is not a particularly developed character. Sohrab is the son of Hassan and Farzana; he was born in Kabul after the couple came to live there and help Rahim Khan take care of Baba's and Amir's estate. His parents are executed by the Taliban and he is placed in an orphanage.

Tragically, he is taken out of the orphanage and used as a kind of sexual slave to Assef, the same man who once raped Sohrab's father. Rahim Kham offers Amir a chance to make up for his past sins against Hassan, and Amir goes back to Afghanistan to rescue Sohrab, his nephew (since he and Hassan are half-brothers). Amir takes Sohrab back to the United States with him, and it is clear that, though things are difficult for him at first, he will have the opportunity, at least, for a fresh start. 

Sohrab represents the best parts of Hassan. When Amir sees the boy for the first time, he says

the resemblance [to Hassan] was breathtaking.

And Sohrab does seem to understand, and tacitly forgive, Baba for not claiming Hassan as his son because Hassan was Hazara. However, he has lived a more difficult life than his father because things are worse for Hazaras (and everyone else, frankly) under Taliban rule. He attempts suicide when it seems Amir is going to abandon him in another orphanage (which is not surprising, given his experience at the last orphanage). 

It is true that Sohrab is virtually silent for a year, obviously recovering from the pain in his life caused by many factors, not just Amir. Despite that, we are left with a sense of hope that he will recover and become a young man who has a chance for a satisfying life. Amir says this about Sohrab's future:

It was only a smile, nothing more. It didn't make everything all right. It didn't make ANYTHING all right. Only a smile. A tiny thing. A leaf in the woods, shaking in the wake of a startled bird's flight. But I'll take it. With open arms. Because when spring comes, it melts the snow one flake at a time, and maybe I just witnessed the first flake melting.

His contribution to the plot is twofold. First, he carries on Hassan's legacy. Second, he is the reason Amir is able to begin to forgive himself for what he did to--and did not do for--Hassan.