This is what Mom always made for us when we were't feeling well. And it is an excellent way to use leftover rice. Best of all, it is simple and fast: Gohan soup. Essentially, it is a miso soup with rice and egg. Here's how Mom always made it for us:
Bring three cups of water to a boil with some sort of dashi stock. I like these fabulous "tea bags" which are essentially kombu (kelp) and bonito in a little baggy that you steep in the hot water. I like them because they are convenient, but they don't contain any MSG. Unfortunately, I've only been able to find them in Japan (I stock up when I visit) and the stuff that is readily available here are the dashi-no-moto granules that DO contain MSG. If you want to avoid that then try makingyour own from scratch. It's actually pretty simple; just boil up a 4 inch-long stip of kombu with a couple tablespoons of bonito flakes and then strain the broth.
Once it has been boiling for a few minutes, reduce the heat to medium-low and add two tablespoons of miso paste (more or less to taste).
Sprinkle in some wakame seaweed (I use a couple tablespoons).
Crack open two eggs and drop it into the soup, then stir very briskly with chopsticks to make it thread-like(like an egg-drop soup).
Finally, remove soup from heat and add a few scoops of cooked rice (this is where leftovers come in handy). The more rice you add, the mushier the soup may be. I usually add ~1-2 cups. I also like to add in some freshly-chopped green onions before serving.
Mayumi loves this - she ate three bowlfuls last time and I felt good that she was getting some wholesome, nutritious food in her belly. Protein, veggies, carbs- it is the best stuff when you're feeling icky (and even when you're feeling fine!).