When I feel especially inspired, I will clean the shavings and sawdust from underneath my lathe, and sweep the shop floor. I generally do this anytime before wife and I go away on a trip - wouldn't want to have a stranger coming into the house and finding that mess.
(that's a variation on the old 'always wear clean underwear so that you won't be embarrassed if you have to go to the hospital in an emergency').
Seriously, in addition to periodic housekeeping, I will occasionally (perhaps once or twice a year) remove the tailstock and banjo, clean the bedways with sandpaper, and apply a coat of white lithium grease. I have tried WD40, and also paste wax, but my experience is that WLG works best for me- it goes on wet and can be rubbed in using a red 3M pad, but then dries fairly quickly, and it doesn't seem to attract sanding dust like other lubricants. WLG is readily available at hardware stores - its the stuff sold for lubricating garage doors.
I will also disassemble the tailstock ram, clean out any dust of shavings that have found their way inside, and use a file to remove the burr where the locking screw rubs against the groove in the ram, before lubing with the same grease.
After cleaning and lubing both the banjo and tailstock, I adjust the nuts that lock them down to the bedways. That said, adjusting the locking nuts, especially the on on the banjo, is something that seems to be needed quite often, so that's probably beyond periodic maintenance and more like an everyday operational check.
I may also remove one of the brushes just to check for excessive wear - on my lathe, it's necessary to remove the motor to get to both brushes, but I find that I can access one brush without disassembling anything, and that gives me a pretty good read on the amount of wear. I've had the lathe for 11 years and have never replaced the brushes - but I have a spare set just in case.
I also will check the belt for tightness. My experience is that the belt will generally let you know when it needs to be replaced, and I do keep a few spares on hand.