Anecdotally, there is a strong correlation between the hanging and the relays clicking away.
Also anecdotally, it seems to be getting better rather than worse (touch wood, cross fingers etc).
About three days ago the scope suddenly completely crashed again while using it, and it failed to ever successfully reboot, stopping at various times during the boot process. After at least half a dozen attempts to get it to boot (turn it off & on, pull the AC cord for a while to let the caps discharge etc), I gave up and put it away in the cupboard to look at later.
A day later, with a view to repairing it, I pulled it back out of the cupboard, and it booted fine. It ran a couple of dozen self-tests fine too. However I noticed that the relays don't click when running a self test. Furthermore, if I set up a semi-complex setup to save in an effort to get the relays clicking, apart from setting the 50/75 ohm termination on a channel, the relays won't click... unless you have the TPP analogue probes plugged in. When doing the self test, you are directed to remove all probes, so the relays aren't cycled.
I plugged the probes back in, and I haven't been able get it to fail again after 1.5 days of solid use.
I now have a new hypothesis... the TekVPI smart probe interface is suffering intermittent connections, leading to a 1-wire comms failure, thus rebooting or crashing the scope. (Previous analysis of these TPP probes show they use a 1-wire solution).
So the next time I get a failure and it won't restart, I'll try rebooting with the probes disconnected.