I take the core out of the sheath and clean both to remove any crud and to inspect the wires that make up the core. If they look reasonable, I wipe the core with a layer of Moly grease (CV joint type) and put the core back in followed by a "little" bit of gear oil (85/90 W). Like George said, you don't want lube forcing its way up into the speedo so a little goes a long way.
BTW, if you can remove the sheath from the car for inspection, that's a good idea too. Cracks and/or chips in the vinyl coating can let water in with obvious bad results. As a minimum, degrease the damaged areas and wrap them with tape. Alternatively, the home centers will have a product called either liquid vinyl or liquid electrical tape that can be applied to the cleaned sheath to provide a patch/cover.