If you slept poorly the night before, you will enter REM faster from a midday nap. Since you're not cycling in and out of REM, the dreams will seem more vivid when you wake up. As far as lucid dreaming, the science on that is sketchy and the personal nature of dreams make them rife for all sorts of New Age nonsense. However, I have found that over years of meditating, my dreams are much more in my control. I've done some pretty cool stuff (like ride a winged tiger through the clouds after it tried to eat me, turning a nightmare into one of the most memorable experiences of my life, awake or dreaming). I spend about 5 minutes at the end of my day breathing deeply and imagining a white screen in my head. This may or may not work for you. Different humans have different paths toward mental clarity. But, I do believe dreams are easier to manage if you spend a few minutes every day clearing out the muck in your brain.

Some people believe a person can train themselves to dream about certain things. I kinda believe that but it might take a lot of effort. I have awakened from a pleasant dream and purposefully went back to sleep to continue the dream and was successful.

Dreaming is uncontrolled thought. On another note, sometimes my dreams are so vivid that I can't discern the difference between a dream and real life. Like many, I have awakened from a dream and actually felt like the dream really happened. If I could spend my whole life in a pleasant dream, I would prefer that to real life (being awake). I think I have had dreams that were so pleasant that the dream was better than any actual experience in my real life.

When I take an afternoon nap, I keep an index card and a pen close by. Occasionally I will dream a good idea and wake up and write it down. I think Einstein did the same thing ( not to equate me with Einstein). Anyway, there is evidence that naps are associated with increased creativeness.