What should I do with the extra grounding wire in my ceiling fan box?
The bare copper wire and any green insulated wires are meant to be connected together, making it a "party of three" as you stated in your question. That said there is another consideration to take into account as well. These safety ground wires are also meant to be connected to a common ground point of the electrical box as well. 

If one of these wires (green or bare copper) coming out of your box already attaches to a safety ground point in the box then you have no more to do than the "party of three" connections.

If neither of these two wires are connected to the electrical box safety ground then it will be necessary to add an additional pigtail green or bare copper wire to the box safety ground and join it with the other three making a "party of four" connection.