Amir's birthday gifts are blood money because his lavish birthday party occurs shortly after his betrayal of Hassan. Hassan was his sacrifice for Baba's approval, and Amir cannot forget his selfishness and cowardice (In Chapter 7, he does nothing while Hassan defends his right to the kite and is subsequently raped by Assef). While Baba most likely would have thrown Amir some kind of party, Amir believes that the huge party--which includes all the guests bringing gifts--would not have occurred if he had not won the kite competition and brought home the last fallen kite to Baba (he only had the kite because of Hassan's loyalty to him). Thus, all of the gifts, Baba's approval, and even the party make Amir feel even guiltier for what he did.
The phrase "blood money" refers to something that one has gained by devious or harmful means. Amir gained Baba's approval from Hassan's literally spilling his own blood for him.