Keep in mind, the speedo cable is most likely turning in a sheath filled with 30 year old grease that has been repeatedly warmed up in the engine compartment. Its probably pretty sticky & gummy.
I would suggest pulling the cable, cleaning it with contact cleaner, and re-lubing it with my favorite Mobil 1 synthetic grease. Don't use a thick grease on a speedo cable, it creates the same problems you have now. A thin grease will liquify when hot and result in no lubrication.
George