Heart rate variability (HRV) is, well, how variable your heart rate is. If your heart beats slow, then fast, then slow, then fast, your HRV is high. If your heart rate is absolutely constant, your HRV is zero. Okay.
It seems to be a sort-of hand-wavey term; when you say "HRV", you might be talking about SDNN (standard deviation of intervals between heartbeats), rMSSD (square root of the mean squared difference of times between successive beats) or any number of other measurements. But these tend to correlate strongly, so it's not unreasonable to just talk about HRV.
In a study by Fang, Huang, Yang, and Tsai (2008), HRV didn't differ between normal patients and insomniacs. Peng, Lin, Sun, and Landis (2007) found that HRV helped their sleep-stage classifier, but they don't say anything about waking HRV. Back in 1973, Volow and Erwin found that HRV had a "marginally significant but unreliable" relationship to drowsiness onset. Furthermore, it's related to many things, but Wikipedia at least says nothing about sleep.
So why do I think that HRV might have something to do with sleepiness? Well, Kaida et al (2007) found that SDNN predicted worse performance on a vigilance task. However, they do note that this goes against some other results, like Hansen et al (2003) and Kohler et al (2006), which showed improved performance with increased HRV. Then there's Tsuchida, Bhuiyan, and Oguri (2009), who threw HRV and facial features into a classifier, finding a correlation with drowsiness of subjects in a driving simulator. ... which doesn't mean that HRV alone is a useful feature.
That's about all I've got for now. So far, HRV does not seem to correlate with sleepiness. It's a complicated measure, and there's just too much going on.
EDIT: it does help determine sleep stage, as Suzuki, Ouchi, Kameyama, and Takahashi (2009) found while combining it with an actigraph-ish watch. Also cool: the fact that they put HRV in a watch.