way back in the old days, there was a dauber in that position.
you would fill the cavity with read lead.
the red lead was used for lubrication purposes when working between centers.
you take a small dab of red lead and insert it into the center-drilled work ends, then place it between dead centers.
since we don't use many dead centers , the practice is fallen from favor
lead is carcinogenic, that too has been a reason for its demise
What does the knob look like if you pull it out and look at the hidden end?
The Atlas machines of the period say to use white lead, which is a liquid or thin paste, with about the consistency of heavy cream. The widget that stores in a hole on the near-side top of the tailstock is called a dauber in the Atlas parts lists. I'm just curious as to whether what Logan called a knob is the original part that functions as what some other manufacturers called a dauber or just a hole plug because they weren't recommending white lead any longer but all of the tailstocks in inventory already had the hole.
Your lucky to have one. Most of them are gone. Mrpete222 on YouTube (tubelcain) suggests using cmd extreme pressure lube in the place of white lead. I found the cmd on Amazon for 10 bucks. It's a lifetime supply. You may be able to sell that piece on Ebay for 40 bucks ;-) I have never seen one in the wild before.
When I studied machine shop in the late 70s, those things were still in place and filled with white lead on the school's WWII vintage lathes. The instructor had squirreled away a 5 gal pail of white lead after it was banned.