How can I refinish small worn areas of a hardwood floor?

https://www.hunker.com/13414843/how-to-refurbish-worn-spots-on-finished-hardwood-floors

Take a photo to the flooring department of the home improvement store and ask for their advice on what products and application materials to buy to get a professional finish.

Take a photo to the flooring department of your home improvement store and ask for their advice on products to buy and application tools to do a professional job. You could try homemade products, but your hardwood is an investment that you need to protect.

try WD-40. Depending on the amount of wear, this may help. I used it on a water stain from a potted plant and it worked. The floor looks just like the original floor. It's been over 2 months and I still can't find where the stain was at.

You just spray some D-40 on the area - I let mine set for a minute - then wipe it clean. It will make your floor really slippery so I just used my usual wood floor cleaner over that area after the WD-40 had dried and it's not slippery.