While I have the 51 year old speedometer cable out of my car...

Should I lubricate it? With what?

Thanks for your help!

Speedometer cable lube. It's a black graphite runny liquid, usually in a very small plastic squeeze bottle or tube. Auto parts store. $4 You say you have it "out". I assume that the outer housing is still in the car, and you just pulled the inner cable out through the top end. Saturate the inner cable with the lube and slide the cable back in, twisting it by hand to feel it engaged inside the transmission. Before you do all this, be sure to inspect every inch of the cable for signs of damage due to rust or wear. Other folks here have other stuff they found that works for them. This is what has always worked for me on all my cars.

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"Louise", a 1976 Dart Custom project, (now sadly reverted to being just an "organ donor" to our other project Darts.)

I've done a few cables in my day. When you spin the inner portion, does it feel like there's crud in it? Does it spin freely? How crusty does trhe whole thing look?
I have just held them up and sprayed the snot outta center section with a Liquid Wrench, PB Blaster, etc. type spray and just let it hang and drip dry. If it spins like there's still dirt in there, bomb it again till it spins freely. Just let hang and drip dry a day or 3, then pour a bit of tranny fluid and let it run through the whole cable. It's a pretty no-brainer type job. Good Luck