Paragraph five of "The Gift of the Magi" evokes the connotation between long names and the upper echelons of society. That is, royalty, professionals, and those belonging to the "upper classes" typically identify with their full names. Especially in the case of royalty, this is to emphasize a family name that the average person associates with power and lofty social status.
At one time, Della's husband was making a comfortable $30 per week. This money granted them a certain lifestyle and social status, for which they had great pride. One manifestation of this pride is using Jim's full name: Mr. James Dillingham Young. Now that the young man makes only $20 a week, they are struggling financially and socially. The family status has declined, which is why the name on the mailbox "seemed too long and important."
However, this same paragraph also points out the fallacy in this line of thinking. The story acknowledges that society generally values money and social status over other things. Della cries because she cannot afford a proper gift. Yet the story observes: "But when Mr. James Dillingham Young entered the furnished rooms, his name became very short indeed. Mrs. James Dillingham Young put her arms warmly about him and called him 'Jim.'" This sets the stage for the moral of this story. It does not matter how much Della's husband makes, or "how long" his name is, because at the end of the day he is her "Jim." Their love for one another supersedes any other material or social consideration. This is the position of the story overall, and the fifth paragraph subtly paves the way for this revelation.