Thanks. That wasn't quite it, but it was one piece of helping me figure out what was going on.
It turns out that, when the machine was imaged and the duplicate PC booted, udev created a duplicate configuration line in /etc/udev/rules.d/70-persistent-net.rules. So, the same exact network interface was in there twice, once named eth0, once named eth1. For whatever reason, only eth1 worked. So, if I manually typed "ifconfig eth1 up" and "dhcpcd eth1," I was fine.
I decided to recreate the image after deleting the 70-persistent-net.rules file on the original machine. That should allow a new file to get written on first boot with the correct eth0 info.