try here for some suggestions
 http://www.corvetteactioncenter.com/forums/c3-technical-performance/58366-speedometer-cable-lube.html

Unfortunately you can't disconnect from the "easy" transmission side - you have to reach up and disconnect from the speedo side.  The clip is fairly self explanatory if you can see it, but I don't remember exactly how it disconnects.  I think it's just a squeeze and pull operation.  Once disconnected, you can pull the cable out of it's jacket.  I think just about any grease would work, I probably used just regular chassis grease or maybe white lithium.  BUT....I think I'd recommend replacing it instead of going to this trouble just to grease.  They are not expensive, especially if you just replace the cable and not the jacket.  But I would recommend buying a quality, correct length from one of the vendors.  My first try was a non-specific part from the Help section of a parts house - not a good idea.     

I just went through this with my 78. I purchased the grease adapter from Mid-America Motorworks to grease it from the transmission side. They recommend white lithium grease and that is what I used. I had to treat it twice as I did not want to over grease it and ruin the speedo. The results are great. My cable was screaming and the needle was bouncing. All issue are now fixed and I am very pleased.

This seems to be the simplest fix without tearing out the dash and risk getting grease in the interior of your Vette. Also there is a "C" clip on the tranny side that would prevent pulling it out through the dash without removing it first.

Hope this helps!

1962-1974 Corvette Speedometer & Tachometer Cable Lubricator
609-467	1962-74	Corvette Speedometer & Tachometer Cable Lubricator Tech Instructions	$19.99	In Stock

Did you get your anser yet??  Let me know, I just may have to take a drive to your house to see your 81!!  I live in Oak Creek, by the MKE Airport!!