What is the best solution for removing the oils of decades of fingers

Murphy's Oil Soap!

Murphys oil soap. try a small with a Magic eraser ( mr. clean).

Yep, Murphys all the way, clean with a magic eraser, wipe with a soft cloth...repeat, repeat..:)

I like lemon oil I think it works well

if the item is truly an antique, be careful about removing the patina. I would not recommend a magic eraser as it it too rough.

Oil dissolves oil -- Coconut oil and a terry-cloth rag, soft tooth brush to get into the swirls, etc. Let it set on the wood for a while -- if it is oily to the point of being able to scrape it up with your thumb nail, it may take several applications. But it's worth the effort.

you can also use olive or almond oil, but coconut is less expensive