Gatsby has two purposes in mind when he calls Nick 'old sport'. First, he is attempting to manufacture the sense of closeness with Nick. Everything that Gatsby does is for Daisy, including his befriending of Nick. Since Nick is Daisy's cousin, Gatsby reasons he will have a better chance at winning her over if he becomes friends with Nick. Gatsby has been shrouding his true identity in lies for years. He is used to not trusting anyone, and as a result, it is difficult for him to create a genuine bond with someone. By using a nickname for Nick, he hopes to engender a sense of trust that is not really there. Secondly, using the nickname 'old sport' makes Gatsby sound like he belongs to old money. This is a turn of phrase that is traditionally used by the wealthy and the educated. It plays into Gatsby's persona. He wants to present himself as a cultured, rich socialite. Everything about Gatsby is an affection, something calculated to impress. He has been acting this part for so long, he may not even realize he is doing it. However, the nickname is still a form of manipulation, whether it is intentional or not.
Hope this helps! Sorry its a lot though. =^-^=