Based on my guess above, this question may answered by the following steps, based on the transfer of spin angular momentum to orbital angular momentum and also the conservation of total angular momentum:
Firstly, you provide angular momentum to the coin by twisting it with your fingers and it has enough angular momentum to remain upright.
It basically stays in that one spot, revolving around the spin axis until it start to lose it's angular momentum through friction with the air and the ground.
As it slows down, the spin axis will start to precess,i.e. to move in a circle, swinging further and further away from the original spin axis until it's bottom pivot point starts to shift on the ground and move away from the original pivot point.
The coin has started with almostly totally spin based angular momentum, but this will be transformed, as it topples over, into orbital angular momentum, which will cause it to roll around the floor.
The coin will then rotate in a spiral shape, possibly starting off in almost completely circular "orbit", but it will move inwards to eventually, as you say, revolve around nothing.
It is the same efffect as a gyroscope undergoes, with the difference that a gyroscrope has a point pivot, so instead of actually moving in a circle, it drops to the floor, the gyro frame spins around a few times, and stops.
Because the coin can undergo rolling motion, it travels around the floor in a circle/spiral instead.
One idea you could try, because I can't at the moment find any web based footage, is to film the coin and then, if you can run the clip slowly, you may get a better idea of the process in action. YouTube should have something like this, somewhere.
Found It: