In the story "A Worn Path," what is it about Phoenix's characterization that makes her such an appropriate person for this role?"
In "A Worn Path," Phoenix is a character coping and surviving as a poor person pitted against indifferent social forces. No one in particular is out to get her, society is just indifferent. She ends up receiving the medicine for free at the clinic, for instance, but not without having to suffer some mild abuse. She is given the medicine out of obligation, but the person in the clinic certainly doesn't feel for her or understand her plight.
The story demonstrates the existence of a person in Phoenix's position. She is an everyday hero, if you will. She is not particularly intelligent or gifted in any way, she just perseveres, she survives. Her perseverance is admirable. The narrator's characterization of her demonstrates what an average person must be like just to maintain the status quo of her existence. She doesn't go to such lengths to acquire a cure for her grandson, but only to keep him alive and breathing. This is existence, according to the story. And Phoenix possesses the kind of personality one must have to survive such an existence.
Consider the mythology of the Phoenix bird. The Phoenix dies, consumes itself in fire, and then rises again from its own ashes to soar again. The grandmother in the story, Phoenix, is a manifestation of the bird. She faces several trials on her travel into the city to obtain the medicine needed for her grandson, but she faces those challenges with dignity, bravery, and grace. She perseveres in the face of the rough terrain, nature, and the man that threaten her because she is determined to do what she must for her family. Grandmothers traditionally are a source of wisdom and strength, and Phoenix certainly displays those characteristics.