What is the "worn path"? Why is the story named after it?
The metaphoric path that elderly Phoenix Jackson treads is long established. She is a woman who has toiled throughout her long life, and the journey she takes in the story is common to many women: she is looking after a child—in this case, her grandson.
The path has deeper symbolism because Phoenix is African American, elderly, and remembers the end of the Civil War. She is first patronized, then menaced, by a white man with dogs, a familiar narrative in a country with a history of slavery. He pulls a gun on her, but she is unmoved, clearly having endured this type of affront in the past. When he drops a nickel, she patiently waits for an opportunity to retrieve it, saying nothing, because she has learned guile as a survival technique.
Near the story's end, Phoenix is patronized again when she is called "Grandma" and subjected to humiliation by being called a charity case in the clinic. She accepts not only the medicine for her grandson, but also a nickel offered patronizingly.
The ritual humiliations and hardships that Phoenix suffers symbolize the deeply-entrenched path that other women and African Americans have suffered before her.