One issue that occurred to me was - did the hard drive need to be defragmented?
On our newer computers, this is programmed to occur at regular intervals. However, this wasn't occurring on the old computer, so I set the de-frag utility to work.
Over 6 hours later the process was still running - on a single partitition that only had a little over 25GB of data! However, the bar that appeared did suggest that the data was highly fragmented. I got various messages about progress: a lot of the time the procedure was "de-fragging" the drive but it also said that it was moving files and later compacting files. At other times, it didn't even tell me what it was doing, but the hard drive was still hard at work, It might have done some other things as well, but I had better things to do than to sit and watch the progress reports that appeared on the screen!
Even so, initial impressions are that all this effort hasn't helped. The hard drive still chugs intermittently and responses can be very slow indeed.
I've also tried to check various other possibilities, such as whether automatic indexing is running - but all have drawn a blank.
I was once told that hard drives rarely fail without warning, so my theory remains that there is indeed an issue with the hard drive. Perhaps the time taken for the de-frag utility to run is itself a further symptom that the hard drive isn't in good shape. Hence my intention is not to stress the old hard drive further by more investigatations and to use it only if it turns out that there's something stored on it that we find we need.