Thanks for these suggestions. I'll drive over to their place this weekend and try those four steps.
Incidentally, the reason for the slightly odd "once only" wording of the title is that they claim that if they reboot the suspect Mac, they can download their emails once, but not subsequently until they reboot.
This suggests something funny with Mail, but (and I've seen this with my own eyes) Safari is blocked as well.
I'm not convinced they are reporting this bit correctly, although I can definitely confirm that the symptoms described in my original email are as described.
