If the revolver is made this way....
....hold the pistol up with the hammer cocked as though you were aiming it.
Now if you look closer at the back of the frame just forward of the wrist of the hammer you are very likely looking at the back of the arbor as it comes through the frame (Some revolvers are designed that the hole for the arbor does not go all the way through the frame.)
If you can see the end of the arbor, (It should form a circular pattern in the frame.) look very closely at about the twelve o'clock position on that little circular pattern which is the end of the arbor. If your arbor is pinned as I am describing, the pin will be something less than 1/8 inch in diameter and it is driven into the frame in the same orientation as the arbor. The best way to understand it is that it acts like a woodruff key on a shaft. Its design is to keep the arbor from turning out of the frame. But the damage which generally causes the arbor to become loose is that the threads on the arbor and in the frame fail. They essentially sheer off at least partially. This is especially true of brass frame revolvers with loose arbors.
If you have a good drill press and if the frame is steel, you can fairly easily drill the pin with a larger hole and just force a replacement pin into the new hole. Three problems with this method:
1. The drill bit can wander off of the existing pin and consequently the new hole does not properly engage both the frame and the arbor. This is especially true in brass frames because the brass is much softer than the steel of the arbor. Drill it slow, use cutting oil, keep and eye on it. As an alternative you can drill a completely new hole and just leave the existing pin alone. Still you have to be careful about the location of the hole. Also you have to be careful about the size of the hole. You are looking at an "interference fit" and these are finnicky.
2. It is very difficult to hold the frame in a drill press vice or jig because there are not many parallel surfaces to clamp onto.
3. The location and alignment of the arbor is relatively critical to proper operation of the pistol. If you fix the arbor but don't ensure good positioning, either the gap can be thrown off or the cycling of the pistol can be loose or tight. To remedy this, take the cylinder out of the revolver but put the barrel back on. Here you are using the barrel to support the end of the arbor and hold it in correct alignment.
Since it is just a wall hanger, I think I would simply leave it alone.