I have a recommendation for a lubricating oil for fans that may be a bit unconventional. I use Boeshield T-9, which is also great as a rust inhibitor. It doesn’t gunk up or attract dirt and dust like other oils, and it seems to stay in place for a long time.
This is a response to Matthias Wandel’s video “Fixing a seized oscillating fan motor”.
I have not been compensated for the makers of Boeshield T9 for this suggestion.