Debian is the name of the base Linux distribution that LinuxBBQ uses. The name is derived from Debby and Ian, who, apparently, started the whole thing. Therefore it should be pronounced 'Deb-ian'. However, you can also just pronounce it 'Deebian' like a normal person (=_^)
The variant that LinuxBBQ releases are based on is the Sid branch. This is the Unstable branch, named after Sid in Toy Story, the evil kid that broke all the toys. This means that you should not be careless when running the BBQ. It is, after all, an open fire that should be maintained lest your house burns down (and on top of that, your meat will be all black and filthy).
With LinuxBBQ releases being Debian-based, this means you can get a lot of useful information from the Debian Wiki.