Bring Ponchos, one for your stroller too if you're bringing one. We went on a rainy December day a few years ago. The lines were super short as people just weren't out. Most people were wearing ponchos and stayed pretty dry. It was our last day, so we didn't feel like forking over the money for Disney ponchos. We were pretty good at staying dry until we went on Luigi's Tires (now closed).
Plan on riding inside rides and seeing shows when it's raining. Then take any chance when it breaks to hit the outdoor rides. There are quite a few indoor rides. And hey, hit up Splash or GRR on your way out of the park. At that point it doesn't matter if you get wet. Enjoy! The crowds will be nicer, even if the weather is miserable.