Any ideas what the original finish would have been on the wood for an early 1950's 640? The wood appears to be Beech.

I'm looking to refresh the wood on mine.

It is beech and looks like they had a coat of thin cheap varnish on them. Not important; replace it with a modern finish; I use a mixture of urethane and tung oil sold by Brownells.
Wipe it down with thinner or xylene, or similar, to remove the grease.

What would you use to strip it?

Any liquid or paste stripper found in a paint store would do it. I like Citristrip because it doesn't dry out and will go thru several coats of finish. Plus it smells like oranges!

Or use xylene like I do. Removes everything including your fingerprints and it smells like kerosene on steroids. But I use it by the gallon for painting military jeeps.