Graphite in a spray can. Can be found in Lowe's or Home Depot. Just did mine a couple of months ago and had good results.
Any Lithium-Based Grease will work best. Always clean the cable with solvent and blow dry (or air dry) it, then add the new Lithium grease.
Grease eventually hardens and the speedo will start to jump around, go to a parts store and ask for 30ND motor oil {non detergent} 30 weight, saturate the inner cable when assembling it. I use 30 ND because any multi visc oil 10-40 will dry up because it has detergent in it.
O'Reilly's Auto Parts actually sells a SPEEDOMETER CABLE lubicant. It comes in a small tube ... costs about $2.50 or so and I just bought some a few days ago for the speedometer in my 32 roadster. After 15 years on the road ... the cable has a slight twitch and need lube. I plan to take care of that item tomorrow
Powdered Graphite comes in a nice little tube like a toothpaste tube with a nice small opening to put into the housing. Never dries out or leaks.
The graphite lube that I used made for speedo cables came in a tube mixed with alcohol. The alcohol carried the powdered graphite all the way to the other end of the cable. The alcohol evaporates leaving the dry graphite inside to lubricate the cable. Disconnect both ends and pour it in from the top. When it comes out the bottom end, you know that the entire length got lubricated.
graphite is what we always used when i worked at service stations. (remember when service stations did lube and oil changes?)
They have been using Grafite ever since Car's were Built so I like Grafite & have been using it on All my cars since the 50's Just my 3.5 cents
If you can get the cable and housing off the car, do so. Pull the cable out of the housing and clean it as best you can. Suspend the housing so that it hangs straight up and down with a coffee can underneath. Now pour some STP down the housing and let it drip into the coffee can. This will take a while. Reassemble. Enjoy. From an old Model A guy.
Many years with motorcycles taught me to NOT use anything but graphite on cables. Oils and greases get very stiff in cold weather. I use graphite spray with the alcohol carrier. The alcohol evaporates leaving only the graphite.
They actually used to sell ( and maybe still do?) a spray in most auto parts called cable lube and that is what it was or is, a graphite lubricant in an aerosol can. grease is cool unless you live where it gets colder than a witches tit then it is not so good.
61Bone and Beaner are right on. Any lubricant will work as long as it hasn't gotten contaminated with dirt or water, but graphite doesn't harden in the cold. That's why its also used on door locks.