What favor does does Nick do for Gatsby?
Nick allows Gatsby to use his house as the meeting point for reuniting with Daisy. Gatsby, however, doesn't ask Nick directly but goes to elaborate lengths to arrange for the favor. He first takes Nick to lunch in New York City, and on the way there mentions to him that Jordan will be making a request on Gatsby's behalf at tea with Nick that afternoon. At the lunch, Nick mentions that he doesn't "like mysteries" and doesn't understand why Gatsby won't come out "frankly" and ask what he wants. Nevertheless, Nick must wait until he sees Jordan to find out the nature of the request.
At tea, Jordan tells Nick the history of Daisy and Gatsby and that Gatsby bought his mansion in order to be across the bay from Daisy. Jordan then asks, on Gatsby's behalf, if Nick will invite Daisy over some afternoon and let Gatsby come by as well.
"The modesty of the demand shook me," Nick records. We note he understands it as a "demand," not a favor. He questions Jordan as to why Gatsby felt he, Nick, needed to be filled in on the entire story of his history with Daisy. He wonders why Jordan doesn't just arrange for Daisy and Gatsby to meet over lunch in the city. Nick learns that Gatsby wants to show Daisy his house, which he can only do if the meeting takes place at Nick's home next door.
Nick agrees and agrees as well not to tell Daisy ahead of time that Gatsby will drop in. The whole episode opens Gatsby up. Nick suddenly sees him now in a new light:
He came alive to me, delivered suddenly from the womb of his purposeless splendor.
Gatsby asks Jordan to ask Nick a favor. He would like for Nick to invite his cousin, Daisy, over for tea, and then have Gatsby conveniently drop by and run into her. This is because he has built his entire mansion and earned his fortune solely to catch her attention and win her affection. He is especially nervous about this meeting, even sending over some of his groundskeepers to mow the grass, trim the hedges, and prepare the house for her arrival. He sends over flowers and also an expensive tea set for the occasion.
When Daisy arrives, Gatsby shows up. At first they don't speak, but later on they have a discussion. Gatsby is extremely nervous and self critical at first, but then invites her over to his mansion where she is thoroughly impressed, even to the point of crying over the beautiful shirts he has hanging neatly in his closet.