One thing you could do is trace the symbolism related to Phoenix's name. A phoenix is a type of magical bird that, when it reaches old age, bursts into flame and is then reborn from the ashes. There are a number of places in the text where the narrator describes either Phoenix herself or objects with either birdlike or burning imagery.
For example, Phoenix has a "golden color" running underneath her skin and cheeks that are "illumined by a yellow burning under the dark." Thus, she seems imbued with a special kind of life and light, as if she is lit up from within. Further, as she walks through the forest, the pine cones "dropped as light as feathers" around her. She talks about how she walks "Up through pines, [and] down through oaks," just as a literal bird would fly through the trees. As she walks, her journey is marked by encounters with more trees: one across the stream, one under which she sits, dead trees that she passes, and live oaks under which she walks. She is initially alarmed by a scarecrow, but once she realizes its harmlessness, she dances in the corn. She is like a bird.
However, when she is faced with a young hunter, her wrinkles go "into a fierce and different radiation," suggesting both extreme heat and light. Then, when she reaches the city and speaks to the attendant, "the wrinkles in her skin sh[ine] like a bright net." For a moment, Phoenix forgets why she came, but then "there came a flicker and then a flame of comprehension across her face." She even describes her grandson, a little boy whose throat was burned with lye, as "a little bird."
Can you think of some reason that the fire which seems to light Phoenix appears to burn so much brighter when she is around the white hunter, the white attendant, and the nurse in the town? Perhaps the process begins again, like the life of a phoenix does, whenever she goes to town: here, she burns, then she goes home, cares for her grandson, waits until the next time he needs his "soothing medicine," and then she starts out again. He depends on her; only she can help him, and so she does despite the miles and their dangers. It is as though she is renewed each time she completes her flight to the town. Her ability to continue to make this trip in her advanced years seems magical, and she does it out of love.
A good thesis could focus on all the symbolism associated with Phoenix's name and the nearly magical process she repeats again and again, just like the bird whose name she shares.
Assuming that your assignment is to write a literary analysis of Eudora Welty's 1941 short story, you could pick one or two techniques and discuss how the author uses them to deliver "A Worn Path's" theme.
For example, there are many elements of the story that are symbolic. The path that Phoenix Jackson walks is both literal and metaphoric. These two elements combined would provide enough material to write a well-developed literary analysis. A thesis for this type of paper should include the techniques to be discussed and the theme. Take the following, for example:
Utilizing symbolism in "A Worn Path," Eudora Welty explores the discrimination and indignities an elderly woman must face as she travels a familiar path, both literal and figurative, of racism in a Southern society.
For one more example, you could write how the setting of the story is integral to understanding its theme. In that case, a thesis could read like the following:
The rural, Southern setting of "A Worn Path" is an important component in understanding Eudora Welty's thematic aim regarding the inhumanity inherent in race and class division in the American South.
I guess a lot depends on what you want to argue after having read this short story. I suppose the central image is the "worn path" of the title, which Phoenix Jackson treads every year for love of her grandson. What you could focus on is how this story presents us with an archetypal journey where the protagonist has to face several challenges in order to be successful in her quest, that is all based on love. Thus you might use the following thesis statement:
"A Worn Path" presents us with an archetypal journey where the protagonist sets out on her quest for love.
This would allow you to examine the archetypal elements of the story and the nature of the challenges that Phoenix Jackson has to endure. Good luck!