What do you use to lubricate a noisy speedometer cable?

Yes Pull it out and clean it with brake cleaner. When it’s dry I use a small amount of wheel bearing grease. Slide it back in the housing.

don’t use white lithium grease in a spray can, It will dry out and form a crust.

I answered the question”what do you use”. Not what does the book say.
If I were to pull the cable core out, clean it with brake cleaner, then lube it with dry graphite, wouldn't there still be a lot of crud inside the housing? And wouldn't this crud contaminate the newly cleaned and lubed core? Is there any way to clean the inside of the housing prior to reinstalling the cleaned and lubed core? You'd need some type of long, tiny brush with some solvent to do this. Ideas?