OK, so I upgraded Webmin (via Webmin) to the latest version (1.840), or I thought I did. I put a top command into CRON so it would run every 15 seconds during the period when the crashes seem to occur (0800 - 0900). At 0814 webmin crashed and the top log showed that copy, bzip2, http and apt-get became active at 0813, using up to 70% CPU, but not a lot of MEM, compared to the 2 java apps. Some kind of install or update had been started. var/log/messages showed this at 0814:

Jun  6 08:14:05 fileserver cron-apt: CRON-APT RUN [/etc/cron-apt/config]: Tue Jun  6 08:03:01 BST 2017
Jun  6 08:14:05 fileserver cron-apt: CRON-APT SLEEP: 604, Tue Jun  6 08:13:05 BST 2017
Jun  6 08:14:05 fileserver cron-apt: CRON-APT ACTION: 5-install
Jun  6 08:14:05 fileserver cron-apt: CRON-APT LINE: /usr/bin/apt-get -o quiet=1 dist-upgrade -q -y -o APT::Get::Show-Upgraded=true -o Dir::Etc::sourcelist=/etc/apt/sources.list.d/security.sources.list -o Dir::Etc::sourceparts=nonexistent -o DPkg::Options::=--force-confdef -o DPkg::Options::=--force-confold
Jun  6 08:14:05 fileserver cron-apt: Setting up keyboard-configuration (1.123) ...
Jun  6 08:14:05 fileserver cron-apt: update-rc.d: warning: start and stop actions are no longer supported; falling back to defaults
Jun  6 08:14:05 fileserver cron-apt: insserv: warning: script 'bubbleupnpserver' missing LSB tags and overrides
Jun  6 08:14:05 fileserver cron-apt: insserv: There is a loop between service monit and bubbleupnpserver if stopped
Jun  6 08:14:05 fileserver cron-apt: insserv:  loop involving service bubbleupnpserver at depth 2
Jun  6 08:14:05 fileserver cron-apt: insserv:  loop involving service monit at depth 1
Jun  6 08:14:05 fileserver cron-apt: insserv: Stopping bubbleupnpserver depends on monit and therefore on system facility `$all' which can not be true!
Jun  6 08:14:05 fileserver cron-apt: insserv: exiting now without changing boot order!
Jun  6 08:14:05 fileserver cron-apt: update-rc.d: error: insserv rejected the script header
Jun  6 08:14:05 fileserver cron-apt: dpkg: error processing package keyboard-configuration (--configure):
Jun  6 08:14:05 fileserver cron-apt:  subprocess installed post-installation script returned error exit status 1
Jun  6 08:14:05 fileserver cron-apt: Setting up samba (2:4.2.14+dfsg-0+deb8u6) ...
Jun  6 08:14:05 fileserver cron-apt: insserv: warning: script 'bubbleupnpserver' missing LSB tags and overrides
Jun  6 08:14:05 fileserver cron-apt: insserv: There is a loop between service monit and bubbleupnpserver if stopped
Jun  6 08:14:05 fileserver cron-apt: insserv:  loop involving service bubbleupnpserver at depth 2
Jun  6 08:14:05 fileserver cron-apt: insserv:  loop involving service monit at depth 1
Jun  6 08:14:05 fileserver cron-apt: insserv: Stopping bubbleupnpserver depends on monit and therefore on system facility `$all' which can not be true!
Jun  6 08:14:05 fileserver cron-apt: insserv: exiting now without changing boot order!
Jun  6 08:14:05 fileserver cron-apt: update-rc.d: error: insserv rejected the script header
Jun  6 08:14:05 fileserver cron-apt: dpkg: error processing package samba (--configure):
Jun  6 08:14:05 fileserver cron-apt:  subprocess installed post-installation script returned error exit status 1
Jun  6 08:14:05 fileserver cron-apt: Setting up webmin (1.831-turnkey+0) ...
Jun  6 08:14:05 fileserver cron-apt: update-rc.d: warning: start and stop actions are no longer supported; falling back to defaults
Jun  6 08:14:05 fileserver cron-apt: insserv: warning: script 'bubbleupnpserver' missing LSB tags and overrides
Jun  6 08:14:05 fileserver cron-apt: insserv: There is a loop between service monit and bubbleupnpserver if stopped
Jun  6 08:14:05 fileserver cron-apt: insserv:  loop involving service bubbleupnpserver at depth 2
Jun  6 08:14:05 fileserver cron-apt: insserv:  loop involving service monit at depth 1
Jun  6 08:14:05 fileserver cron-apt: insserv: Stopping bubbleupnpserver depends on monit and therefore on system facility `$all' which can not be true!
Jun  6 08:14:05 fileserver cron-apt: insserv: exiting now without changing boot order!
Jun  6 08:14:05 fileserver cron-apt: update-rc.d: error: insserv rejected the script header
Jun  6 08:14:05 fileserver cron-apt: dpkg: error processing package webmin (--configure):
Jun  6 08:14:05 fileserver cron-apt:  subprocess installed post-installation script returned error exit status 1
Jun  6 08:14:05 fileserver cron-apt: Setting up rpcbind (0.2.1-6+deb8u2) ...
Jun  6 08:14:05 fileserver cron-apt: insserv: warning: script 'bubbleupnpserver' missing LSB tags and overrides
Jun  6 08:14:05 fileserver cron-apt: insserv: There is a loop between service monit and bubbleupnpserver if stopped
Jun  6 08:14:05 fileserver cron-apt: insserv:  loop involving service bubbleupnpserver at depth 2
Jun  6 08:14:05 fileserver cron-apt: insserv:  loop involving service monit at depth 1
Jun  6 08:14:05 fileserver cron-apt: insserv: Stopping bubbleupnpserver depends on monit and therefore on system facility `$all' which can not be true!
Jun  6 08:14:05 fileserver cron-apt: insserv: exiting now without changing boot order!
Jun  6 08:14:05 fileserver cron-apt: update-rc.d: error: insserv rejected the script header
Jun  6 08:14:05 fileserver cron-apt: dpkg: error processing package rpcbind (--configure):
Jun  6 08:14:05 fileserver cron-apt:  subprocess installed post-installation script returned error exit status 1
Jun  6 08:14:05 fileserver cron-apt: dpkg: dependency problems prevent configuration of webmin-filemin:
Jun  6 08:14:05 fileserver cron-apt:  webmin-filemin depends on webmin (>= 1.831); however:
Jun  6 08:14:05 fileserver cron-apt:   Package webmin is not configured yet.
Jun  6 08:14:05 fileserver cron-apt: 
Jun  6 08:14:05 fileserver cron-apt: dpkg: error processing package webmin-filemin (--configure):
Jun  6 08:14:05 fileserver cron-apt:  dependency problems - leaving unconfigured
Jun  6 08:14:05 fileserver cron-apt: dpkg: dependency problems prevent configuration of webmin-time:
Jun  6 08:14:05 fileserver cron-apt:  webmin-time depends on webmin (>= 1.831); however:
Jun  6 08:14:05 fileserver cron-apt:   Package webmin is not configured yet.
Jun  6 08:14:05 fileserver cron-apt: 
Jun  6 08:14:05 fileserver cron-apt: dpkg: error processing package webmin-time (--configure):
Jun  6 08:14:05 fileserver cron-apt:  dependency problems - leaving unconfigured
Jun  6 08:14:05 fileserver cron-apt: dpkg: dependency problems prevent configuration of xserver-xorg-core:
Jun  6 08:14:05 fileserver cron-apt:  xserver-xorg-core depends on keyboard-configuration; however:
Jun  6 08:14:05 fileserver cron-apt:   Package keyboard-configuration is not configured yet.
Jun  6 08:14:05 fileserver cron-apt: 
Jun  6 08:14:05 fileserver cron-apt: dpkg: error processing package xserver-xorg-core (--configure):
Jun  6 08:14:05 fileserver cron-apt:  dependency problems - leaving unconfigured
Jun  6 08:14:05 fileserver cron-apt: dpkg: dependency problems prevent configuration of xserver-xorg-video-intel:
Jun  6 08:14:05 fileserver cron-apt:  xserver-xorg-video-intel depends on xorg-video-abi-18; however:
Jun  6 08:14:05 fileserver cron-apt:   Package xorg-video-abi-18 is not installed.
Jun  6 08:14:05 fileserver cron-apt:   Package xserver-xorg-core which provides xorg-video-abi-18 is not configured yet.
Jun  6 08:14:05 fileserver cron-apt:  xserver-xorg-video-intel depends on xserver-xorg-core (>= 2:1.15.99.903); however:
Jun  6 08:14:05 fileserver cron-apt:   Package xserver-xorg-core is not configured yet.
Jun  6 08:14:05 fileserver cron-apt: 
Jun  6 08:14:05 fileserver cron-apt: dpkg: error processing package xserver-xorg-video-intel (--configure):
Jun  6 08:14:05 fileserver cron-apt:  dependency problems - leaving unconfigured
Jun  6 08:14:05 fileserver cron-apt: Errors were encountered while processing:
Jun  6 08:14:05 fileserver cron-apt:  keyboard-configuration
Jun  6 08:14:05 fileserver cron-apt:  samba
Jun  6 08:14:05 fileserver cron-apt:  webmin
Jun  6 08:14:05 fileserver cron-apt:  rpcbind
Jun  6 08:14:05 fileserver cron-apt:  webmin-filemin
Jun  6 08:14:05 fileserver cron-apt:  webmin-time
Jun  6 08:14:05 fileserver cron-apt:  xserver-xorg-core
Jun  6 08:14:05 fileserver cron-apt:  xserver-xorg-video-intel
Jun  6 08:14:05 fileserver cron-apt: [master f632e22] committing changes in /etc after apt run
Jun  6 08:14:05 fileserver cron-apt:  30 files changed, 207 insertions(+), 212 deletions(-)
Jun  6 08:14:05 fileserver cron-apt:  rewrite webmin/uninstall.sh (69%)
Jun  6 08:14:05 fileserver cron-apt: E: Sub-process /usr/bin/dpkg returned an error code (1)

And today webmin seems to have reverted to 1.831. For info, bubbleupnpserver is one of the 2 java apps I have installed, xserver-xorg-video-intel is something I installed to make the video work properly on the eeePC. The CRON command which kicks all this off (5-install) looks like this:-

autoclean -q -y
dist-upgrade -q -y -o APT::Get::Show-Upgraded=true -o Dir::Etc::sourcelist=/etc/apt/sources.list.d/security.sources.list -o Dir::Etc::sourceparts=nonexistent -o DPkg::Options::=--force-confdef -o DPkg::Options::=--force-confold

Which is beyond me. So, some kind of update seems to be kicked off, which includes webmin, as a result of which webmin crashes. There seem to be a lot of error conditions encountered, but I'm not sure if any/which of them are serious. I am considering reinstalling everything in stages to see at what stage this starts to happen, but would prefer a more analytical approach, if anyone can decode the above messages.
Marmite.