A couple of other amplifications:
1) Yes, I'm sure it's the fan and not a short in the board. That is indeed a common cause of PSU clicking, but not in this case: When I physically spin the fan with my finger, I can hear that there is something in there clicking around. Moreover, obstructing the fan with a pencil prevents the noise from every occurring.
2) Oiling the fan bearing with a drop of mineral oil did exactly nothing.