I proved a topology theorem in a dream once.

Before I went to sleep, my inability to prove it had been bugging me all day long, and I suspected it'd be featured on the next morning's (way too early) final exam for my university course. I solved it in my dream, woke up, wrote on my whiteboard what I remembered and sure enough, it was correct. I worked it a few more times to cram it into my memory before running to my exam.

To my great delight, the ability to prove that theorem was featured heavily in one of the exam's questions, and helped me do quite well on the exam overall.