Galvanic skin response (GSR), AKA skin conductance/conductivity (and a lot of other names), might be a useful signal. It's cheap and easy to measure and widely known. It generally increases with arousal, and therefore is used in lie detectors. And E-Meters. Okay, maybe those are not great examples.
Davies and Krkovic (1964) found that EEG, skin conductance, and performance in a vigilance task all correlated for 10 college students. Skip forward 30 years, and Lim et al (1996) found out more details about the EEG-skin conductance (SCL) relationship. They found a correlation between each brain wave band/position pair and SCL, but then after searching for possible covariates (I might be botching this concept entirely) they came up with a model that only correlated Beta 3 (18-25Hz), Alpha 1 (8-10Hz), and SCL. I mean, the model was:
SCL = -0.685*EEG_Beta3 - 0.045*EEG_Alpha1 + 9.556
(both waves measured at Fz, which is the center of the frontal lobe). Generally, EEG and SCL both correlate with physiological arousal decline. Is that the same as sleepiness?
Bundele and Banerjee (2009) found that they can pretty well distinguish between pre-driving and post-driving skin conductance readings from drivers. They say that the difference is fatigue, so skin conductivity is a signal for fatigue. I say, could be fatigue, or could be stress or any number of other things that rise when you drive. Still, it's something.
Daniel Kramer (2007) studied performance on a video game and found that it correlated with skin conductivity. And Shimomura et al (2008) found that some analysis of skin conductivity correlated with task difficulty. These seem not really relevant here. But now I'm thinking, performance is linked to high arousal, and high arousal is linked to conductivity. If we just say "conductivity is about equal to arousal" and measure your arousal all day, maybe that would be higher on days where you're less sleepy.
So... what? Does GSR measure sleepiness? Probably not in any one case; if you measure your GSR right now, it's unlikely to say "GSR = 0.93, therefore you're sleepy." But maybe over time it'd be worth it.
EDIT: a couple other papers that I only have abstracts of:
Yamamoto and Isshiki (1992): well this looks promising. "We selected the GSR as a physiological index that indicates the awake level."
McDonald, Johnson, and Hord (1964): "Results showed that there were no differences between groups in GSR; however, the drowsy group showed consistently fewer spontaneous GSRs". (I guess GSRs happen in spikes, not just as a value; so maybe more awake people get more spikes in GSR)
Scholander (1961): response amplitudes of electrodermal activity was influenced by sleep deprivation.