Many mornings and some middle-of-the-nights I wake up with stream-of-consciousness thoughts. Most of the time the thoughts are next-steps or even solutions to problems that I'd been working on the day before. Other times, they're introspections; wonderful and very welcome views of my "me" that are otherwise obscured by the normalness of daily life.
To learn more about "sleeping on a problem", read Bob Morris's review of the most excellent book The Net and the Butterfly: The Art and Practice of Breakthrough Thinking by Olivia Fox Cabane and Judah Pollack.
Below are some of the trappings of those "me" introspections.