Checking the logs in the AlertMe dashboard showed that the problem was appearing whenever the configuration was being fetched. The message would read "Configuration load failed" and the Hub would sit in a semi-dormant state with the LEDs set to red. Sometimes it would attempt a reload, which would always result in the same error, and sometimes all of the lights would go out and the Hub would appear to be dead.
It turned out that the solution to this problem was very easy! I changed the original power adapter from the Hub for a new one; I happened to have a nicely over-spec 5V 2A power supply with the correctly sized barrel plug on the end. If you're not certain about this, any 5V power adapter should do the job, so long as it's capable of supplying 0.5A or more.
Just a word of warning; never connect a power adapter that has a higher (or lower) voltage than 5V, even if the plug fits the socket. You will fry your Hub doing that. Any current rating above about 0.5A should be fine (usually available adapters are 0.5A, 1A, 1.5A, 2A).
Why Was This Happening?
It seems that it's during the configuration load that the internal radios for communicating via Zigbee and the mobile network are powered on. This obviously has a fairly reasonable power draw on the unit and obviously my power adapter had degraded and wasn't able to take the load any more. The battery wasn't able to take the strain either, maybe because it's also reaching the end of its life, but very possibly because the adapter hasn't been able to charge it. The circuits would by low on power, causing a fault condition that was either caught by the software (which then put the Hub into a waiting mode) or just knocked the whole thing out of whack.
So, if you're having this problem, buy yourself a good quality 5V 1A switch mode power supply and give it a try. I will report back on whether my system stays happy!