Grace is the name of the narrator's young daughter, who dies of polio when she is two years old. Her death marks a turning point in the narrator's relationship with Sonny; it is on the day of her funeral that he writes to Sonny and reestablishes contact.
Grace's name is symbolic because it is through grace that the narrator and his brother are brought together. But it is also symbolic of the suffering both men endure. The narrator is critical of Sonny because he sees him as weak and views his drug addiction as a kind of character flaw. He thinks of himself as someone who has "made it" and sees his life as a school teacher and family man as proof. This makes it difficult for him to empathize with Sonny or his troubles.
The death of Grace, however, provides the grace the narrator needed to begin to understand his brother. The childhood Grace never was able to live symbolizes both the bleakness of life in Harlem and a "clean slate," the kind of new expression and life made possible by overcoming suffering. It is only by enduring his own heartbreak that the narrator finds the grace to listen to Sonny, both in the apartment, where Sonny tries to explain how life has been for him, and at the club, where Sonny's music does the talking.