I'm of course talking about nocturnal penile tumescence (NPT), often experienced as a waking (and perhaps unexplained) erection. However, the stiffening in question is not the work of devilish succubi or experimenting aliens (both sometimes share the blame). Rather, a healthy male experiences up to five of these a night as part of his natural REM sleep.
In fact, NPTs are so normal that doctors test for them when diagnosing erectile dysfunction. If these erections occur on their own during the night, then the patient's problem is likely psychological rather than physiological.
But why do they happen at all? Since they occur during the REM cycle, there's always the potential for a sexually-charged dream tie in, but it's not necessary. The main NPT theories actually have nothing to do with dreams. Rather, the body may be oxygenating the penile tissue as a sort of maintenance procedure -- and the stimulant for this is thought to be the release of nitric oxide by nerve fibers in the penis.
Oh, and then there's the whole full-bladder angle. The two often occur together and scientists have taken note. So some theories view morning wood as the body's anti-bed wetting measure, or perhaps a wake-up call to the sleeper himself.
So in other words, gentlemen, a little morning wood (or even a lot of it) is nothing to worry about.