I use AGS White Lithium Grease for lubricating (among other things) any auto-inject floppy drives I encounter, and it seems to work quite well (when used in moderation, of course; if I use too much, it actually tends to make things worse, especially when it's cold).

You can get 35.4g squeeze tubes at a hardware store (in the US; don't know about elsewhere).

Good luck on that drive!

c