/sbin/chromeos_shutdown has this snippet:

...logic to unmount all the stateful things...
if [ $rc -ne 0 ] ; then
  mount > /mnt/stateful_partition/shutdown_stateful_umount_failure
fi

# Just in case something didn't unmount properly above.
sync
...

we should be logging an UMA stat for this event.  looking at cs/, the only thing that checks shutdown_stateful_umount_failure is the platform_CleanShutdown autotest.

strawman design:
have some startup code (send-boot-metrics.conf maybe?) check for this file.  if it exists, log a new UMA stat, cat the file into the syslog (so it shows up in feedback reports), and delete the file.