In "A Good Man Is Hard to Find," how does the grandmother's violent encounter with the Misfit lead her to Christ?
The grandmother is very self-absorbed and preoccupied with herself. It's her decision to go on vacation, and where they go, and that they drive to the house she wants to see. It's essentially her fault they are in the fix they are in, and she is the one who recognizes the Misfit. Had she not said she knew who he was, they may have gotten off without being shot in the head and left in the woods.
It is only when the grandmother is faced with her own mortality and the death of her entire family that she has a thought which is unselfish. Her talk with the Misfit begins as a selfish attempt to save her life and the lives of her family, but then ends up with the revelation that the Misfit is "one of her own children"--meaning that he is like all other people. He was not destined to be the evil man people think him to be...events in his life could have lead him anywhere...just like events have led the grandmother and her family to the same isolated road where the Misfit and his crew happen to be.
The Misfit responds with the sentiment that if the grandmother had been in such a serious situation (a gun to her head) all her life that she might have been a better, more tolerable person.
So, it is her desperate attempt to save her own life that leads her to redemption--however brief it is, it is significant to her character development.