It looks to me like part of the dash lights dimmer switch, or another control that rotates.

 

do your wipers work?

 

It is the copper disc from inside the ignition switch. Likely has been replaced over the years when it fell apart or was broken apart for missing keys. Trash now. You would know if yours was missing.

 

Now that would make sense. Because everything ,except for my clock, on the dash works as far as I can tell. Light switch, dash dimmer, wipers, wiper fluid motor, and heater blower switch and heater flaps like defrost, etc. all seem to work just fine.
However, it does appear that the ignition may be new because the keys match the type of keys that came from Dennis Carpenter when I got new door locks.
I was thinking it may have been part of the levers with the selector bracket that move the cables to open heater flaps.

 

Tell us about your fan switch cleanup.

 

Once you remove the knob you have to remove the nut on the back side that holds the switch onto the heater switch bracket. Once I got in on the bench i cleaned all the male connectors and then I gently bent the tabs up that are holding the brown board, with the connections, and separated the switch. Inside is just a copper plate looking piece that slides up and down and makes contact to the brown circuit board connections. There is spring underneath that applies pressure to the copper plate and a plastic housing that the spring sits in. So all i did was clean it really good and scuff up the connections with light sandpaper and reapply electric grease. Then reassembled it all. Its really basic.
from there I tested the switch and resistor on the bench with a test light and a 12v batt.
following the wire diagram of the car I knew that the switch receives power from the fan and grounds through the resistor to the switch.
Hope this helps!
the pic of the switch is just for reference, it is not the actual one for a 64 galaxie. Its the best pic I could find of a similar switch.
the metal housing may be a little different from car to car, but the functionality was pretty much the same as far as I can tell through most of these old cars in the early 60s, with a 3 speed blower switch. Please correct me if I am wrong.

 

The fan switch in my '64 now I took out of my dad's '61 wagon when I first got my car. Now my fan just suddenly quit working but think this time it's the motor not switch as it worked that morning then that night it didn't when turning it on. Have not got to it to see how things test. Not looking forward to removing the heater box to get to it. Tried removing the cover on the firewall but couldn't find the wires. I only need it for defrosting the windshield mostly so that's when i actually miss it. Thanks for sharing the little diagram, that makes things easier.

 

Have you checked the fuse?

 

Yes I have, that was the first thing. Tried cleaning the contacts in the box too.

I did best I could reaching up behind to see that the wires are still connected to the switch also. I haven't gotten to seeing if the resister on the heater box is ok or not yet.
I know it controls the speed of the fan or thinking that's part of it's purpose but if completely trash would the fan still work just no speed control or would it be dead. Somewhere i have one (in a safe place i could find later) for it I took from another car that was in the junkyard that was like new. It may never be seen again til I don't need it though as I have no idea where that safe place is!!

 

do check out the resistor block. it's easy to get to and remove. i had one lose the screw that held the contact on it