What do you use to polish wood plinth?

I have an early model SOTA Star and the wood finish is dull in one spot. It's Koa wood and I've been using lemon oil and it restore the luster in the wood for awhile. Then a week or so later the plinth looks fine except in that one spot. Is there something I could use that would be better and restore that spot?
Contact a local woodworker or furniture maker. I think you are looking for a wax instead of an oil. It sounds like you have a spot where the finish has worn off. That is why the spot does not match the rest of the plinth.

You need to find out what the top finish had on it that has worn away. Was it dull or shiny? Clear-coat, varnish?? (maybe a wax or a sealant of some kind, urethane or polyurethane)? JAWAG!
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Agree with the others. Wood can be finished in many ways with many different materials. Oil? Shellac? Varnish? Wax? Polyurethane? Matte? Satin? Gloss?

Attempting to refinish one part of an older wood piece without knowing anything about the original finish is guesswork. With so many possibilities, most guesses will be wrong and some may cause problems that can't be fixed.

Your results make it evident that lemon oil is one of many possible wrong guesses, so stop applying it. Ask SOTA what the original finish was. If they can't help, ask a local wood finishing expert. They need to be local so they can actually see and touch the wood. I know everything on the internet is true, but no one can deduce your turntable's wood finish from a distance. ;-)
I would agree with Elizabeth that someone tried to fix a marking and ruined the topcoat in the process. One solution is to remove everything from the plinth so that you are left with just the wood to deal with. Take 0000 steel wool and remove the balance of the to topcoat to an even dull finish. Then remove all dust with tack cloth and re-apply a topcoat finish of your choosing. Try to pick a satin or a semi-gloss to help conceal any imperfections that may exist. Apply as many coats as you wish till you reach the look you are most happy with. Just repeat the steel wool and tack cloth between each coat. I would say that a spray poly would be the easiest to work with or you may want to use a waterbased clear finish. But from your description, I think refinishing is the inevidable solution. Trying to patch it in will create more frustration and may cause further damage, requiring restaining and then refinishing. You can find many woodworking sites that will walk you thru these steps to make it seem less daunting.
If Howard's Feed-N-Wax didn't work I would just adhere a nice Hello Kitty sticker to the spot and call it a day.
Do not use steel wool with water-based finishes. The fine metal dust (even after using a tack cloth) will oxidize.