Instructions
Go to any town on the mission map and select "Pub" from the list of places you want to visit. It's the first option above "Shop" and "Monster Bank."
Speak to the bartender inside the pub and then select "Missions" from the menu. Use the d-pad to scroll through the list of missions until "Gold Gil" is highlighted and press "A." The mission description will display. For this mission, the shopkeeper wants you to investigate the origins of the golden gil in his shop. The bartender will inform you that this is a "Dispatch" mission and you must select a team member to send out.
Press "Down" on the d-pad to scroll through a list of all the members you have in your party. A picture of the party member displays on the right, along with his level. Highlight one of your party members who has a picture showing him jumping up and down then press "A." Doing so is almost a guarantee that he will complete the mission successfully. A character with a portrait of him walking has a lower chance of success and one that is kneeling has almost no chance.
Press "Left" on the d-pad to highlight "Yes" when asked if you accept these conditions and then press "A." The game will subtract 1200 Gil from your coffers and the character you have chosen to do the Golden Gil mission will be gone for twenty days. The days display on the bottom right of the screen when you are moving around the game map.
Walk around on the game map in order to pass the time. Move the cursor with the d-pad until a town is highlighted then press "A" to move your party there. While they are walking, the counter at the bottom right of the screen showing the days will indicate how much time is passing. After 20 days, a report window will appear showing "Mission Cleared" and "Dispatched Unit Returned." The character you sent on the mission will return to your party and you will get ancient coins, 6400 Gil and one random item for successfully completing the mission.