Jay Gatsby imitates the habit of the upper-class of British young men who refer to friends as "old sport" so that it will seem that he has attended Oxford University.
In his effort to make it look like he has reached a social level equal to that of Daisy Buchanan, Jay Gatz has recreated himself as Jay Gatsby, whose resumé includes having graduated from the prestigious British university at Oxford. In truth, Jay Gatz only attended Oxford a few months. This fact is revealed in Chapter Seven while Jordan, Daisy, Tom, and Gatsby are in the New York hotel. As they drink cocktails, Tom confronts Gatsby about his pretense of having graduated from Oxford: “By the way, Mr. Gatsby, I understand you’re an Oxford man.” Uncomfortably, Gatsby replies, "Not exactly." Tom then insinuates that Gatsby has completely fabricated his having attended Oxford. “I told you I went there,” said Gatsby. Tom answers him, “I heard you, but I’d like to know when.” Gatsby clarifies,
“It was in nineteen-nineteen, I only stayed five months. That’s why I can’t really call myself an Oxford man....It was an opportunity they gave to some of the officers after the Armistice,” he continued. “We could go to any of the universities in England or France.”
When he hears this statement of Gatsby's, Nick narrates that he wants to get up and pat him on the back. "I had one of those renewals of complete faith in him that I’d experienced before" (Ch.7). Nick is relieved to witness Gatsby's honesty because he has been disturbed by Gatsby's pretense in other situations.
The repeated use of this annoying speech habit is part of Gatsby's affectations. Nick is annoyed by the phrase and we can see this habit of speech as part of what makes Nick distrusting of Gatsby initially. Nick does not trust Gatsby, in part, because he does not like him and he does not like him, in part, because Gatsby constantly calls him "old sport".
Gatsby's use of this phrase demonstrates a selfish or myopic demeanor in him. Gatsby is not genuine with Nick and so uses a generic and generically "chummy" term to express his mild affections for his neighbor.
Also, we can see that Gatsby wants Nick to like him. In the baldness of his intentions and the poor way in which they are carried out, we see that Gatsby is not socially adept, even if he is glamorous, powerful, and successful.
The repetition of this term from Gatsby serves to emphasize the importance of Gatsby's appearance to others, besides Nick. His unoriginal use of language feeds into the perception that his is inauthentic.