What is the cheapest way I can monitor my brain activity?

That's a tough question to answer.  Unfortunately brain monitoring technology isn't necessarily a hot topic for R&D the way other technology is, most. equipment is large and in labs and costs alot of money.  The problem is that its very hard to tell exactly whats going on as so many areas of the brain activate differently for different reasons or stimuli.  The simplest ones now are available in a Star Wars concentration game in which you attach one EEG sensor to your head.  Although that one isn't very efficacious for your desired activity. Luckily, I do believe there is one EEG device on the market that measures brain waves and it doesn't break the bank. Check out their website http://www.neurosky.com/ Of course, if you live near a state university with a med school or research facilities of the such you might well be able to borrow higher end equipment like an EEG net with like 64 individual sensors, but that would also require alot more work to fully use. Actually reviewing it again the Mindwave by neurosky costs 100 dollars and is designed for exactly what you hope to do

I'm a fool when it comes to technology. So I'd suggest that every morning, you write down all the details you can remember about the dream you had the day before. It may be a bit tricky as dreams are usually fragmented. Try to write as much and as quick as possible. Sometimes you'll only need keywords instead of full sentences.