Challenge coins come in a variety of types and styles, did you have a specific style in mind or have you seen a coin on one of the manufactures sites that you think would be a good direction for what you need?
Have you guys set a budget for the coins themselves? This is important since after you decide how many you need made the remaining budget would determine what options are available for the production of the coin. Size, 1 sided vs. 2 sided, color, material, edge type, and finish etc.
Have you decided on what copy is to go on the coin? Is certain imagery that needs to go on the coin? If not, is there relevant imagery that would represent the Office of the Mayor? (logo, architecture, flag, mascot/animal, statue, person etc.)