To think about the meaning of a name, start first with the literal meaning, and then second with the associated or metaphorical meaning. To "weather" something is to make it through; the expression is usually used in relation to getting through tough times, as in "The family weathered the Great Depression." "All" means just that-- all. Everything. Together with her title, her name would mean she's an old woman who has been through everything and survived, or who will make it through everything. Since the story follows Granny as she reviews her memories, we can see that it definitely applies. She's done everything. After all, this is a woman who fenced 100 acres by herself!
The word "weather" means that she has experienced a lot of things and he have endured a lot of hardships and sufferings through the passage of time, to make it through thick and thin and to conquer all odds. It is trying to create a impression that the grandmother had suffered through hard times and have overcome many obstacles to what she is now. The word "all" means that he had gone through many bad and tragic times and she have seen everything so she is immune to them now. Overall, it is trying to depict the grandmother as a strong-willed character, who had play an important part in the family affairs and her grandchildren lives. Example was when she was very young, she was jilted and abandoned by her former boyfriend, George at the altar, and from this tragic experience, she pick up herself and overcome from the setback and married another man. It shows that she is a determined person and she has the "never-say-die" positive attitude.