This week I learned more about Apple's Time Machine. My wife's iMac needed to have its motherboard replaced, due to the capacitor plague. Kudos to Apple for extending the warranty to cover this issue, as it meant that we didn't have to pay for the replacement. Co-incidentally, a Grundig HDTV Satellite tuner failed recently due to the same problem (I opened the case, and saw the signs of the capacitor leakage.)
One of the consequences of the replacement of the motherboard is that the MAC address of the network card has changed. And this means that the external USB drive being used for the Time Machine backup was no longer recognized, since it appears that Time Machine embeds the MAC address in the drive identifier for the backup archive.
To resolve, there is probably some way to edit the MAC address, but I didn't bother. Instead, I noted that the USB drive was still using the Master Boot Record (MBR), therefore I decided to re-partition the drive with the Apple Partition Map, which is best used with the PowerPC-based iMac. I then used the "Change Disk..." option under the Time Machine panel of System Preferences, and started a new full backup, which corrected the problem, at the expense of some older backups.