Update: Yup, it's rats. Couple little (kinda gross, fair warning) stories...
First, I went around the entire chicken run/base of the coop and reinforced the mesh wire. I was determined to make it impenetrable. I attached new underground wire, topped it with crushed stone and heavy rocks, and packed it down. A couple nights went by...every morning I checked for any signs of digging. There were a couple attempts on the outside, but no more rats were getting into the run. No more tunnels. After a couple days, I stopped finding any attempts at all. Had they simply given up?
I have two indoor cats. A few days after making those adjustments to the run, I awoke to find a dried blood smear on the hardwood floor in our living room along with a small, rodent-sized kidney organ.

My cats are skilled hunters; they have caught mice in our house before and for some reason, when they eat them, they always leave behind the kidney.

This looked exactly like a mouse kidney, but bigger- about the size of a quarter. It seemed that one rat had somehow made its way into our house, but clearly didn't last too long with our cats. Maybe the rats had turned on our house in search of food and shelter after being blocked from getting into the run?
We have two 30-gallon buckets right up against our house in the backyard. We'd had a few big rain storms recently, so the buckets were filled with water, almost to the brim. A couple days after discovering the rodent-sized kidney, my husband discovered two drowned rats in one of the buckets. It looked like they had been in there for at least a couple days. So, that explained the sudden halt in signs of digging. It seems that the rats have defeated themselves, making my job much easier.
Anyways, that confirms the culprit behind the tunneling. Rats, just like many of you suspected. It has been about a month since discovering the drowned rats, and I've seen no signs of them since.
But this morning I went out to the coop to let the chickens out in the morning and, lo and behold, there was a big old rat snooping around in front of the coop (not inside the run.) As soon as it saw me, it scuttled off into the woods and disappeared. So, apparently they're still around. But the good thing is they have not been able to get back into the run (or the house.) Our coop is pretty solid, so they've never been able to make it into there either.