(The remainder is mostly a note to myself, so that the next time around I'll know what to do when things break.)
Since there was a kernel update waiting to take effect, it was a good idea to have updated the kernel-dev packages as well, just in case. In fact, this time it was indeed the case: on rebooting (after 333 days, sigh!) the graphics card drivers wouldn't work with the new kernel.
Fixing the video: Figure out the graphics card model (NVIDIA Quadro NVS 285), and download the relevant software from the NVIDIA website. Run it and, like it did this time, it might want to compile some modules against your spanking new kernel. Let it do as it pleases. Things may not look quite OK even after this (I tried running xscreensaver-demo using a GL* screensaver to confirm -- and got a bunch of "GLX" errors). Check/edit the /etc/X11/xorg.conf file, and ensure that in the Module section Load glx is there but Load dri is commented out. The instructions I was following also had me edit the Device section to read Driver "nvidia" instead of Driver "nv" (so that, I suppose, the driver installed by Nvidia's script is used; not sure why that script didn't edit that into the conf file). Quite unexpectedly, at this point, everything started working.
Fixing the wiki: This is why I rebooted the server in the first place. The wiki on the server was a little broken, thanks to mysterious SELinux issues. It was probably a consequence of software updates leaving the newly installed files in wrong domains. I thought rebooting will fix all that. It didn't. What did fix it was the fixfiles relabel command.
And today's was the last lecture for the semester.