Why do you think the author might name the protongonist after this mythical creature ?
The Phoenix is a mythological creature who, every 1,000, is consumed with fire, only to regenerate from the ashes stronger than before. This is the picture that Welty wants us to connect to Phoenix Jackson. In the first few paragraphs of the story, we see the physical description of Phoenix -- she is older, her eye sight is failing her, and she isn't as strong as she once was. However, her spirit is strong, energetic, and determined. Because of this determination, she is able to rise above her physical challenges to make this journey to town again and again. Thus she is a type of a Phoenix, rising about her challenges to be stronger and more energetic for the next journey.