An inspiring topic. So much is possible that is rarely tried.
Randy, you can drill a hole from the mouth to the back of the solid head, and thread a light cable (heavy fishing line works) with a ring on the finger side and fasten the other to the jaw in any number of clever ways as suggested by others above.
This makes it a "cable puppet", where unlike a marionette whose strings rise up visibly, the cable puppet strings thread through its body, more like living creatures. The cable can be as heavy as a bicycle cable, or as light as various fishing line, threaded through a "sheath". Flexible fishing line holds up well, and the sheath can be slippery teflon "spaghetti" from a ham radio store. Teflon allows for great subtlety. The jaw can be weighted or spring-loaded with elastic, so that you need only pull the cable. Add a second cable and you can make the rat's eyes open wide, or maybe a grin. Dual-cables can counter-pull to and fro, useful for eye directions.
I recommend to check out
"Supermarionation" (click for info) for some admirably evolved examples. It mentions Henson company uses comparable advanced methods.
If you're making videos from your puppets, it's good to know about
"Go Motion" (click for info) animation as well, smart techniques I saw applied at ILM and LFL years ago. Here's something from robotics that puppet designers can use:
Nitinol wire changes form with temperature (be sure to get LOW temperature). Here's one interesting demonstration (I do not work for them) . Click picture to visit source:
The hand isn't strong - it's a demo. Someone handy with prototyping could easily convert an audio signal to varying DC (as with a VU meter) and thus a small puppet's mouth move as you speak into a microphone. The same method could make a bird flutter exactly with sounds from a bird recording, or a zillion other imaginative uses.