I finally found information regarding the rebuilding of the kernel on the BBB (Beagle Bone Black) - turns out its a Debian Wheezy distribution. The basic information on doing this is at http://eewiki.net/display/linuxonarm/BeagleBone+Black. The instructions pretty much matched reality (with alterations for the directory structure I'm doing my development in).
Note that, as of this writing, I have not tested the build kernel - I can't seem to find all the MicroSD chips I used to have. Sigh.
In the image, the BBB is to the left of the keyboard below the screen - you might be able to see one of the blue lights as it flashes. The screen has the following windows on it: * Terminal window - this is logged into the BBB and shows the stock OS version and the tail end of a hierarchical process list. * Several web browser windows that are opened to the BBB on the desk, the eewiki web site, and the BeagleBone site. At the moment, these are my primary reference sites.
My next task is to try to boot this (stock) kernel to ensure all the development processes work. Then I'll start work on the device driver.