Good question. The grandmother does seem to change at the very end of the story, though it doesn't change her destiny in the story at all. Throughout the story, the Grandmother is very selfish and, frankly, annoying. She pushes everyone around, purposely antagonizing her son and his wife. The detour was her idea, though she never takes responsibility, and the situation that the family finds themselves in is because of her faulty memory. Even while her family is being systematically executed, she is still looking out for herself, trying to falter the Misfit into letting her live. The instance of change, while brief, happens after she and the Misfit question the resurrection of Jesus. Seeing his human side for the first time, she finally realizes her own selfishness and hypocrisy, and reaches for him in a moment of sympathy. That moment is her last, as he shoots her three times as a response to her action. The glimpse of change is present, but her life is taken before any indication of real, permanent change.
Her religious epiphany at the story's end provides the philosophical thrust behind the narrative. She is selfish and pushy; in fact, her desire to see a house from her childhood results in the family's death at the end of the story. She demonstrates racist behavior and she reveals a superior moral attitude.
The Misfit's explanation for his behavior provides an opportunity for the self-centered Grandmother to reflect on her beliefs in the moments before he shoots her "three times through the chest." In her final moment, the Grandmother reaches out and touches the Misfit, whispering "You're one of my own children!" The Misfit's final commentary on the Grandmother is that "she would of been a good woman ... if it had been somebody there to shoot her every minute of her life."