Why does the author repeatedly refer to Della and Jim as "the James Dillingham Youngs?"
The name "Dillingham" has money and upper-class connotations. Jim's last name is Young. The "Dillingham" must have been a sort of tribute to Jim's grandfather, who was evidently an important man, as symbolized by the big gold watch. Jim could never have afforded to buy such a watch himself. It had been handed down from grandfather, to father, to Jim.
Now, there were two possessions of the James Dillingham Youngs in which they both took a mighty pride. One was Jim's gold watch that had been his father's and his grandfather's....
Della not only loves her husband, but she is proud of being married to a man who comes from impressive background. This explains why she is so strongly motivated to buy him an expensive fob that will suitably complement his watch. She knows he is proud of it, not only because of its beauty and value, but because it is a symbol of his ancestry. She herself is evidently from an inferior background. O. Henry hints at this in her dialogue. For example:
“Don't you like me just as well, anyhow? I'm me without my hair, ain't I?”
Jim would never use "ain't," nor would he ever correct her for using the word herself. He is a gentleman. He suffers more than she does from their privations. She has multiple reasons for wanting to buy him a beautiful Christmas present. She isn't doing it merely for love. She feels sorry for him because he has to work so hard at a low-paying clerical job and because he has to support her. There is a strong suggestion that she may be pregnant. When he comes home, the author records some of her observations.
The door opened and Jim stepped in and closed it. He looked thin and very serious. Poor fellow, he was only twenty-two—and to be burdened with a family! He needed a new overcoat and he was without gloves.
Jim also has to use an ugly attachment for his gold watch.
Grand as the watch was, he sometimes looked at it on the sly on account of the old leather strap that he used in place of a chain.
In buying Jim a fob that will be "worthy" of his watch, Della is trying to be "worthy" of her distinguished husband. Her strongest motivation is her fear of losing him. She likes being Mrs. James Dillingham Young. But when she sacrifices her hair to buy the platinum fob, she fears she may be in greater danger of losing him.
The name "Dillingham," reflects Jim's superior social background, his present fallen condition, his pride in his ancestry, his possession of such an expensive watch, his distinction in the eyes of his adoring wife, her motivation to buy him an exceptional Christmas present, her sacrifice of her beautiful hair, and his sacrifice of his watch to buy her the combs which she may not need for years.
O. Henry suggests early in the story that Jim may also sacrifice the name Dillingham along with the watch.
Also appertaining thereunto was a card bearing the name “Mr. James Dillingham Young.” The “Dillingham” had been flung to the breeze during a former period of prosperity when its possessor was being paid $30 per week. Now, when the income was shrunk to $20, the letters of “Dillingham” looked blurred, as though they were thinking seriously of contracting to a modest and unassuming D.
When the economy improves and Jim can afford to buy new calling cards, the cards will read "James D. Young." He will have merged into the lower middle class psychologically along with Della. Both were inordinately proud of their "treasures." In sacrificing her hair and in Jim's sacrificing his "Dillingham," they are both accepting the reality of their humble stations in life. They are somewhat similar to Monsieur and Madame Loisel in Guy de Maupassant's famous story "The Necklace," in that they have to face the reality of being plain, ordinary people.