What does the grandmother say to the Misfit to try to convince him not to kill her? Does the conversation alter his viewpoint in any way?

The grandmother tries to appeal to the Misfit's nonexistent sense of morals by claiming to see in him his "good blood" and by asserting that she "knows" he "come[s] from nice people." She delivers these appeals in a panic, desperate in her knowledge that she is likely not going to survive these moments. After all, by this point in the story, her family have not survived this encounter with the Misfit and his gang, so her last resort is to encourage the Misfit to get in touch with his good side and allow her to live. She even says to him, "You're one of my own children!" as a way to get some sympathy, as she believes that if she identifies with a mother figure, surely the Misfit with show her some mercy.
These attempts to get some mercy from the Misfit fail utterly. The Misfit clearly has no regard for mothers nor for the children of mothers. Not only is the Misfit unconvinced and unaltered in his viewpoint, he is scornful and mocking of the Grandmother's last gasps. The story finishes with these lines, leaving the reader with no uncertainty as to the Misfit's absolute lack of emotion and morality:

"She would of been a good woman," The Misfit said, "if it had been somebody there to shoot her every minute of her life.'"
"Some fun!" Bobby Lee said.
"Shut up, Bobby Lee," the Misfit said. "It's no real pleasure in life."

In an attempt to convince the Misfit not to kill her, the grandmother repeatedly asserts that the Misfit is "a good man" and that he isn't "common." The irony of this statement is that the Misfit is clearly an outlaw bent on murdering the entire family, and the grandmother's statement that he's "a good man" does nothing to stop his murderous rampage or change his evil intentions. Indeed, as the grandmother continues to assert that the Misfit is good, the criminal casually orders his henchmen to begin killing the grandmother's family. As such, the grandmother's statements seem to be her attempt to convince herself that she is not about to be killed, despite mounting evidence to the contrary. These attempts grow increasingly pathetic, as it becomes clear that the Misfit is determined to stick to his original plan and kill the grandmother and her entire family.