Speedometer needle bounce, 1968 GT350

[1967 eight barrel]

NO LITHIUM GREASE!
https://www.macsautoparts.com/ford_falcon_mercury_comet/speedometer-cable-lubricant-special-graphite-type-1-2-fluid-oz-382709.html

[rhjanes]

A bounce like that can also signal that the inner cable is weakened.  Mine was doing that.  I took it apart and found the inner cable starting to come apart.
What I do, and I also do this on my 1970 428 with manual choke, for the manual choke cable.
Remove the entire cable.  Take it apart.  Use brake cleaner down in the outer part to get out all the old, who knows what lubricants.
Use some brake cleaner (and gloves) on some rags or paper towels to clean up the inner part (or just get a replacement).  Inspect it.
Get graphite from your local big-box hardware store (ask for it over in the lock area).  "puff" that graphite into both ends of the outer part.  Use some compressed air to blow it down into the cable.  Put some graphite on another paper towel and work the inner part thru that.  Assemble together and onto the car.
Go for a drive.