Looking to replace my 9 year old desktop.  Should I go new or refurbished or open box? see below?

If it were me, I would go new over the rest unless you get one Hell of a killer deal.  Any computer that was used, you run the risk of the prior user doing things like overclocking the processor, which wears it out faster or not cooling it properly and then you never know what kinds of changes they did to the Bios, which you can default back to the standard setup, but they could have changed fan speeds, overclocked, or anything else that would change the way the parts work and then put it back before selling it.  If it were my initial computer, I would always go new.  If you are buying as a second or backup computer, it is all about the lowest price vs. getting what you want in it.

Your Budget, Your Choice, BUT

"refurbished" usually means it already broke at least once.

"open box" usually means somebody already decided they didn't like it

"New" is still in the original packaging with original OEM bloatware and the full factory warranty.

At the moment during the current COVID pandemic, prices for computers/components are sky-high, so new or open box is a good idea. As you haven’t stated your location then it’s difficult to give any local recommendations. Dell do an outlet store for refurbs and some firms have an eBay store.

Anything out there is good. 300 plus.   Better to  get  a used 1 year old  dekstop small form.  unless  a refurb comes up that would  be better as they last. Open box are entirely reliant on people no messing around  the computer  itself.

thats up to you