Don't use white lithium grease...it can actually make it worse. use light wheel bearing grease, first pull out the inner cable to see if theres any kinks..if either squared end is slightly bent it can cause a speedo bounce. if the speedo bounces from maybe 10 M.P.H. to 30-35 it's usually a bad cable, if it bounces all the time no matter what speed it more than likely needs a speedo service.
Try Redline gauge works (661)259-8891 ask for Shannon,(I used to work there) they do work for Chip Foose, Jay Leno, Steve Strope of Pure Vsion, and many others.
Any other gauge questions send me a message, I can repair most of the smaller gauges at home fuel, temp, oil press. clocks, circuit boards, voltage limters etc.
Redline can handle speedo repairs tach conversions full restos.