I've bought this cheap mp3 player for a specific purpose, and once I got over how much technology you get for a fiver nowadays (seriously, it's like living in the near-future) I poked around it and found that it had an undocumented feature of playing avi files.

I suspect undocumented as it seems to be very, very picky about what it will play. The glorious web proved oddly quiet on the matter. Luckily, it came with a sample avi that does work.

I broke out avidemux and examined it with the simple properties dialogue. Seemed pretty standard, so I made another avi with the same video codec, image size, frame rate, audio channels, bitrate, codec, sampling frequency; no dice. It is declared to be an unsupported format.

So how, then, can I dig out more information from the working sample avi I have? This is an exercise in academic enquiry and just for the sheer entertainment value of it.

Use ffmpeg and the libxvid codec and it should work. See:
http://draconishinobi.50webs.com/linux/encoding.html

I have tried many other ways to encode it, using mpeg4, mencoder, but this is the only reliable way.