How to fix hardwood floor with wear from an office chair
Before you apply any stain, try dampening the area with mineral spirits (which will evaporate off fairly quickly, unlike water). That will give you some idea of what color the wood will be if you just apply a clear varnish over it. Also remember that oil-based finishes will tend to add some yellow to the hue (as will shellac).  It's possible that you don't actually need to do anything more than giving it a light scuff-sanding and a new coat or three of whatever's already there.

Remember to do something to keep the chair from wearing thru again too quickly... felt pads on the chair legs, an area rug or chair mat, non-marring wheels...