Hadamard, in his book, quotes Poincaré describing a night when he had drunk black coffee and couldn’t sleep: “Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination”. Poincaré was more aware of his unconscious processes than most of us. (This and his stature as a mathematician suggest that probably this awareness is a good thing).

On this analogy, the time spent on preparation (thinking about the problem) gets the “blocks” of ideas sorted out so that we know their shape well. The unconscious incubation phase involves this sort of trying out of combinations. Probably some of this happens while we are asleep; the brain forgets the combinations that didn’t work and only remembers the ones that did. (Interesting philosophical question there: can we forget or remember something that never reached our awareness? I would say yes.) The illumination phase then occurs when you sit down to think about the problem, and find that one idea “brings with it” another which fits and advances your understanding. The random dreams would be mostly the combinations that didn’t work, and it is important to the process that these should not be remembered.

So remembering your dreams may be a good way to train your awareness, but is probably not a good way to do mathematics!

More evidence today to back up what you are saying about the link between dreaming and learning:

http://news.bbc.co.uk/1/hi/health/8638551.stm

Now we just have to figure out how to get ourselves to dream about our research! Perhaps compulsory beds-in-offices is the way forward?