Try wiggling the lever to see if it stops. Mine used to start clicking the flasher for no reason too and wiggling the lever would stop it. I cleaned the switch with compressed air and stopped the problem. I don't think I even removed the wheel. 

My money's on something in the steering column. You could probably unplug the column plug from the harness and it would stop. I suspect either the switch behind the steering wheel is filthy or so worn out that two contacts are touching. Needs cleaning or replacing, is my bet.
If you had tilt, it could be worn wires in the tube now touching, but tilt is pretty rare in a 61. I've had numerous wiring issues with brake and turn signal lights in tilt columns. If I get one now, I set it in one spot and never move it.
I always think it's going to be easy fixing wires in the column, but somehow it never is. I quite often get annoyed and end up swapping the whole column and using or selling the other for parts.

bjburnout wrote:

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The 61 signals are operated by a Bowden cable that moves a switch located on the mast jacket.
This device is self centering but after many years of use they quite often got dirty and would stick in
the closed position resulting in your problem, the cable can easily slip if the switch is dirty or broken.

Locate the switch on the mast jacket under the dash, it is attached by 2 screws.
Detach the switch from the column you can now operate it by hand for testing........
These are available in reproduction.

cheers
bj

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 Ah yes, the steel rod. I forgot about those older ones. I had the same type in my 60 Impala, I believe. I think I swapped columns on that, too, just because there was other damage.