Eudora Welty's "A Worn Path" follows the journey of Phoenix Jackson, an old African-American woman who must travel on a dangerous rural road to seek out medicine for her grandson, who had accidentally consumed lye and is at risk of having permanent throat damage due to the swelling.
It it, thus, tremendously appropriate that this character be named "Phoenix" like the mythical creature who (as you have mentioned) bursts into flames after five hundred years and is then re-born from its own ashes. Phoenix Jackson must face tremendous obstacles in her quest for the medicine, from the physical pain of making such a journey at her age to the external threats of the environment (wild animals, dangerous entrapments of the landscape, unbearable heat, etc.).
Phoenix's love for her grandson provides her with the fiery "re-birth" and strength necessary to carry out this task. Although she almost fails when she experiences a lapse in memory that causes her to forget the reason that she came in the first place, Phoenix experiences "a flame of comprehension" (again, another reference to the symbolic fire that renews her will) and is able to tell the nurse why she requires the medicine. Triumphant, she leaves town with the cure to save the child and plans to purchase a new toy for him.
The mythical creature, The Phoenix, is able to rejuvenate, to rise again from ash and decay, and to continue on past what logically it should. These features make the creature relatable to the character of Phoenix Jackson. She is older, starting to lose her mind and her physical abilities. Yet, despite the fact that she should not able to make this trip to town on her own still, she is able to push on and persevere. When she gets to town, her mental limitations begin once again taking over, and she momentarily forgets why she has come. Just like the phoenix, she rises from this lapse once again with renewed strength, showing that her love and concern for her grandson is stronger than the grasp of age and time.