For some reason most of the men in my life enjoy giving me ratty old knitted items to repair. I don't knit socks for DH and his friends anymore but they still produce various pairs of damaged
mittens and gloves. The pop-up mittens today weren't even handknit but DH asked me to fix the thumb. He has recently started using trekking poles for his hiking and backpacking which caused the hole.
The very first thing I do is handwash the item in question. These mittens have what looks like wood stain on them which didn't go away. Then I estimated on the gauge and needle size and chose some Elann Sock it to Me yarn and size 2 (US) dpns. I like using a yarn with a bit of nylon for this purpose to help the repairs last longer.
I cut off the tip of the thumb and ripped back to a point where the yarn was no longer thin. I put the sts on the needles and started knitting. To determine how long to make the thumb, normally I'd count the rows but my gauge was a bit finer so I added a few rows. The original mitten thumb looked like it was just a few rows of K2TOG for the thumb tip decreases so that is what I did as well.
I'm not sure if this will be a recurring problem but if it is the next time I repair them I might add some duplicate stitch to the area that is prone to holes to help the thumb last longer.