You can remove and inspect the inner cable for damage, but my guess is that the problem is inside the speedo head itself. On my "barn find" TR3, the trip and odometer mechanisms were so stiff that the speedo cable would wind up every time the little ratchet tried to turn the wheel, then unwind while the ratchet was resetting. When I disconnected the trip & odo, the waver went away. Took a couple of iterations, but I finally got them cleaned, polished and lubricated to where the needle doesn't waver today. Of course I also cleaned and lubricated the speedo mechanism.
If you do decide to pull it apart yourself, be very careful when removing the needle. I got a little carried away with one speedo (fortunately one I was just robbing parts from) and the spindle snapped before the needle came loose. Now I twist the needle rather than just pry it up.