Blotchy Wood Finish on Vintage Furniture Pieces  

You don't need to only...
You don't need to only remove (human) oil though, as the color has changed/deteriorated/been removed due to abrasion/wear; you also need to ADD color, right? The finish has gotten super thin in spots. I think you're at the point where you either need a complete refinish or a customer that does not object to the patina of fifty years of touch...

Maybe try...
I'm pretty sure that this happens when the original finish is worn and raw wood is exposed. I've used 0000 steel wool and Howard's Feed-in-wax before and it does seem to help even out the surface. I don't know of a product that will completely fix the issue. Maybe do a test on a small section and see if this helps. Thats my best suggestion other than refinishing. Also I'm not huge fan of Howard's Restor a finish. It looks great for about a day and then seems to dry out again.

Although...
If you just get a finish touch-up marker that's close enough color-wise, I suppose you might daub/blot/blend for a while and end up splitting the difference well enough, adding color where you need it and keeping the vintage color intact. That's probably all I'd do, but I'm lazy-

Do you mean...
... Restore-A-Finish? Because that isn't oil.

Your only best and real option...
... to get the color uniform is to get a liquid stripper/refinisher and remove all the varnish or most-likely lacquer and then to finish as you please.

I was hoping for something...
I was hoping for something that was more of a quick fix, not looking to refinish or strip the piece. Maybe I will just add some stain to the lighter areas and see how that goes, as objectworhship suggested.
Thanks you very much everyone!

I actually like Howard...
I actually like Howard Restor a lot! It does dry out and leaves a powdering substance that requires some wiping with a paper towel but when you have a piece that's 50 years old with a lot of scratches and the wood is really dry looking, it really evens everything out and brings the color back very nicely. As a total amateur and one who's not looking to do any major refinishing work on vintage pieces I find Howard Restor-a-Finish a great product!

If the piece is lacquer...
... which is likely, you can hunt down some Mohawk sprays and 'tone' the color correct, although this is tricky business and could be a disaster. If you do it right, though, the new lacquer will simply dissolve into the old. Hard to do, but a pro could probably do it for you.
Liquid refinisher is a lot better than stripper, though, and I think you should consider it if the unevenness really bothers you. It's serious chemicals, though.

Oy.
It's most likely lacquered. All you have to do is wipe the lacquer off with lacquer thinner or acetone, which you can get at any hardware store.
Then oil as you wish. It will even out very nicely without adding color or using methylene chloride stripper or any of that stuff.
It's not a super quick fix but it's the only one that works in my experience when dealing with a worn lacquer finish.

Thanks Mgee and Spanky!
Thanks Mgee and Spanky!