As George is the name of the man that jilted Granny Weatherall all those years ago, and as this is still something that Granny Weatherall remembers with shame and embarrassment, we can infer that Granny Weatherall wants her dignity back as she feels that this is what primarily was taken from her when she was jilted. Note what she says in her mind about George:
Yes, she had changed her mind after sixty years and she would like to see George. I want you to find George. Find him and be sure to tell him I forgot him. I want him to know I had my husband just the same and my children and my house like any other woman. A good house too and a good husband that I loved and fine children out of him. Better than I hoped for even. Tell him I was given back everything he took away and more.
Note the way that this quote is paradoxical. Granny insists she has forgotten George, yet at the same time feels the need to send him this message, that indicates that she has not forgotten him. The purpose of this message seems to be to prove to George that she has lived life anyway in spite of the hurt she suffered because of his rejection, yet at the same time she is aware of "something not given back," some emptiness in her life because of her jilting. It is perhaps this sense of dignity lost that she wants to regain by the end of her story.