Mrs. Jones wants Roger to remember her as someone who cared about him.
When Mrs. Jones grabs a hold of Roger, she knows his story and what she will do as a response. From the first moment they interact, she is in complete control of the situation. She has sized Roger up and knows exactly what he needs. It is for this reason that "Sweat popped out on the boy’s face and he began to struggle." He knows that he messed with the wrong person.
Mrs. Jones wants to provide an emotional structure for Roger. From directing questions to him, she knows Roger lacks a stable parental figure. It is for this reason that she makes him clean himself and cooks him a meal. When she refers to him as "Son," and forges a relationship with him through talking about her past and his present, Mrs. Jones embraces a form of restorative justice with Roger. She wants him to know that his actions have consequences on human beings. In appealing to his humanity, she is convinced that she can change the way he acts towards others.
As she gives him the money and tells him to "behave himself," Mrs. Jones wants Roger to remember that someone cares about him. She impresses upon him that his dignity is not something to be surrendered so easily. One of her last words to him is that he should "not make the mistake of latching onto my pocketbook nor nobody else’s—because shoes come by devilish like that will burn your feet." Roger is struck with the gesture and wants to say more, but sees the door closed. This shows that Mrs. Jones has accomplished what she set out to do. Roger will remember her as one who showed love and care when few others did.