Longitude certainly is arbitrary, though I would say that latitude is not.<br /><br />We could, for example, create a simple galactic coordinate system that first defines a galactic plane. The galactic center (perhaps the supermassive black hole) would need to be located on this plane. The plane would be constructed such that the standard deviation of distances of stars from the plane in the galaxy would be minimized.<br /><br />We define the origin as the supermassive black hole. We can then start defining axes. The x-axis would be the line from the origin within the plane that passes closest to the sun. (Why not?) Since the sun may not be in the plane, the axis will not pass directly through it.<br />The y-axis is the line normal to the plane passing through the origin. The positive direction could be in either direction. The z-axis is then set so that the three axes form a right-handed cartesian coordinate system.<br /><br />And there you are!<br /><br />Of course, the coordinates of all objects would be continually changing, except for the origin's. The sun would always have a zero y-coordinate, but its x and z coordinates would change.<br /><br />This is just one possible way to do it; there are many.