I have now tested my NUC again a couple of times. If I try to boot it without a display attached, nothing at all gets written in the logs. The booting should be done by elilo, and the linux kernel is passed "vga=normal" as a parameter. For me it's not random: it seems to never boot, unless connected to a display that's turned on.
I bought a HDMI-DVI adapter that plugs in to the NUC. It doesn't help. Likewise, I noticed that plugging a HDMI cable to a projector that's turned on, but using another input, does not help. At least the NUC is very stable and doesn't need to be restarted often..