I managed to fix the problem later on, with a bit of fiddling. I think the solution I came across first is more complicated than a solution I found later, so I'll describe the easy solution first.
Not all input options are enabled in Volume Control (right-click speaker icon in panel, or use the Terminal command gnome-volume-control) by default. By selecting Edit > Preferences, it's possible to enable more options. The process is described in this tutorial, so I won't repeat it, but I'll add a screenshot.
The solution I actually used involved making changes in alsamixer, although I enabled the microphone just by fiddling with the controls until it worked. Using Volume Control is probably more sensible. Here's how to fiddle, if you must:
Select [Capture] with the tab key.I think what worked for me was fiddling with the Input Source options. Here's a screenshot. I'm still not sure what all the option do, and of course the options may differ with different sound cards.
Select options with the left/right arrow keys.
Enable with the space bar.
Change options with the up/down arrow keys.
This advice may be helpful if fiddling doesn't work.
Footnote: When I'd got about half way through writing this, I kicked myself- I'd used the Alsa Volume Control a year or so ago in Ubuntu before Ubuntu switched to the PulseAudio system. I must have enabled my microphone under Alsa in Ubuntu and then completely forgotten how I'd done it- one of the reasons I now write this blog. And in the hope that somebody else in the same situation might find it useful.
Update: Some time after installing Debian Squeeze beta (as was) I noticed that my microphone had stopped working again. I managed to fix it using a tip from the ALSA unofficial Wiki page above.
Comment from reader: In my case, with the snd-hda-intel driver there was a mislabeling of capture channels (probably due to incorrectly selected model in snd-hda-intel). The alsamixer capture item labelled "Digital" was controlling the microphone recording volume, while "Capture" and "Capture1" didn't change anything.I'm not sure whether Squeeze muted the required digital channel or I did it myself.