For those who aren't all that adventurous or just want something simple and dry to do, check out some local pet boutiques and barkeries/bakeries. Since most of the owners are dog-centric people, it's a great place to get info on new or dog-specific toys, food and tons of other stuff.
If you and your pooch are in some serious need of grub, consider hitting up a dog-friendly restaurant or doing a short patio tour. Plenty of places in South Carolina are dog-friendly, but in the rain, you'll just need to find the ones with covered patio areas like the Lost Dog Cafe. It can be a calm and peaceful way to spend a rainy day, as long as it's not too windy, and you and your dog will get to have some delicious food in the process!