My guess is the cable has dried out. If you don't have cruise, you can disconnect the speedo cable from the back of the odometer then pull it back through the firewall. I use graphite and pour down the cable and work the inerits to get it to go down the cable.
You can do this from inside but using graphite is REALLY messy, that stuff gets on everything, a small plastic bag and rubber band around the end before you shove it back through the firewall will help eliminate more graphite from coming out onto your floor board
It may take about 10 to 30 miles for the lube to work down and settle out the bouncing needle. You can pull the guts out of the cable too, and lube it and put it back in, if you do it this way, be sure to disconnect it from the trans so you can be sure the guts are inserted all the way down the cable housing.
I have the same problem but its not too bad yet. I was wondering if you can disconnect it from the lower end and pull out the inner cable to lube it. I dont know if it will feed back up thru the outer casing into the speedo either. If this can be done what would be the best thing to lube it with. I hate having to mess around under the dashboard.
OK, I know the answer to this question.........The cable has to come out from above, the way the cable is made, you can only get it out from the top. You will have to remove the left side AC duct, reach up and depress the clip holding the cable to the speedo, or tach,(for 74 and earlier), then drop it down where you can get a hold of it. Pull the cable out(Inner cable) clean it, lube it, I use white lithium, and I apply the lubricant as I am putting the cable back into the black housing, to help keep down on the mess. Put everything back together and it ought to work, without bounceing...........Good luck...Steve
I had the same problem on my 77. At first I pulled the cable out and cleaned it up...loaded with rust sediment. Added graphite solvent, reinstalled and great for a week or so of casual driving. Problem came right back and also a whispering noise from the speedometer started. Went to chevy dealer, picked up new Delco cable and installed it. Not that a bad job to tackle. Only difficulty was getting under dash to make the speedometer connection.