Many thanks for the suggestions thus far. I love the sound of the hundred-pounds GPSMap 196, but I fear that kind of purchasing opportunity comes along only once in a lifetime.
A moving map isn't essential for me. The old Garmin 55 AVD simply had a CDI and a numerical readout giving distance, G/S and estimated time to destination, which was plenty. If there was any way of prodding the thing back into life at a reasonable cost, I'd gladly do that, but the manufacturer is neither servicing nor supporting it.
For my purposes, I don't see myself going the "iPad + whatever" route. I don't own an iPad or any other tablet. What I'm hoping to find is essentially what I previously had: a light, cheap, simple GPS that will not take up much space in (or add much weight to) my flight bag, and will get me out of trouble using its own independent power source (i.e. battery) if everything else in the cockpit bar the wet compass should go kaputt.
I do think that something with an aviation database is necessary, though. I wouldn't want to have to programme the backup with a series of waypoints every time I go flying, nor -- if the necessity of actually using it arose -- would I fancy doing so with one hand while flying the aircraft with the other.