So I set forth on a quest to acquire software and filters to clean it up. I read up on Temporal and Spatial Smoothing filters, on detecting duplicate frames, on frameserving with AviSynth and VirtualDub. Then, in the postprocessing guide over on Doom9, I find a neat little tip: capture the source more than one time (because the noise doesn't seem to be constant), and average the two captures.
Some tests on the short clip I have suggest that I would need four captures to get anything near decent quality. That's a lot of diskspace (which isn't really a problem because I can always offload some episodes onto DVD+RW) but also a lot of time.
What I really want/need is a filter that compares two clips side by side. If the delta (the change) between consequtive frames is a lot higher in clip A than in clip B, it should get the frame from clip B -- and average the two frames otherwise.
Unfortunately, there is a 5-day waiting period on the Doom9-forums before you can post, so I have to practice some patience before I can ask whether such a filter exists. But perhaps someone knows of such a filter? kees_s? greatbiggary?