A search on amazon turned up some products called "cotton darning thread".

cotton darning thread

Thanks for looking, Donna. I had seen the ones on Amazon. They are not darning thread. It seems like someone in China decided darning was a synonym for sewing. The reviews say these chinese products are all ordinary sewing thread. Darning cotton has a multi-thread strand, like embroidery floss. You can use all 5 or so threads, or separate. If I am unsuccessful, in a pinch I will try embroidery floss, though that is shiny and slippery, and probably does not have good wearing qualities.

Okay. Gotcha. I have only darned socks that I have knit, so I use either the same yarn or a similar yarn. But you are looking for another animal altogether. I know some yarn companies sell a finer yarn that you can knit along with your regular yarn at the heels or toes to prolong wear, but I've never actually seen it, so I can't tell you if that might work.

The only suggestion I can make is to search websites that sell sock yarn. I'll see if I can come up with something.

I found this on ebay:

Darning thread

And this on etsy:

More darning thread

You’re giving me an idea, now. Maybe I should just buy sock
yarn? Where do you get this? I should go look at some in a store and see if the weight
will be right. ButI would guess than handmade socks use heavier yarn than ready
made, right?

You can buy sock yarn at Joann and Michael's, but their selection is limited. I don't know if you knit or crochet, so don't know what your knowledge is about yarn weights. Sock yarn most often sold is a #2 weight, usually seen as a number on a symbol for a skein of yarn on the band of the skein. You can find finer yarn, but I don't like knitting socks with yarn finer than #2 - the grandkids would outgrow them before I got them finished, LOL.

You might have to go to a local yarn store to find finer yarn (#1), but you could at least see if that is what you might be looking for. Of course, LYS's are notorious for overpricing their merchandise, but if you only need it for darning, you won't need much. You could probably order online and pay less, even if you include the shipping costs. There are lots of reasonably priced sock (and finer) yarns online. Try Knitting-Warehouse.com, Knitpicks.com, Yarn.com, even ebay.

Thanks again, Donna. No, I don't knit or crochet. My Mom did, but all just thicker thangs liek afghans.