[Since the word essay has been used rather than story, the assumption will be made that the question refers to an essay entitled the same as Chopin's story.]
The fact that the word the, rather than a is used in this title arouses more curiosity in a reader because it places emphasis upon the word story as a very particular narrative. And, it is the story of just one hour, as opposed to what a newspaper report may call an exciting story: "The story of the hour": that is, the "best story of the last 60 minutes," or "the best story of the most important hour."
Singling out just one hour as opposed to one day also raises questions in the reader's mind, such as "What is so important that happened in this particular hour?" Interestingly, the time is never mentioned in the actual narrative. Chopin simply begins with "Knowing that Mrs. Mallard was afflicted with a heart trouble," and ends with "...had died of heart disease....," thus lending the narrative a temporal sense.