A thin film of very light oil. If you have some SAE 10 or 20 oil for the lathe spindle, or other light oil around, that's what you want. You generally wipe on the oil from a small rag that is lightly oiled, but not quite saturated. When you get the chuck assembled, stand out of the way when you first spin it up, as it will likely throw off some oil.
You want to wipe oil on all the sliding surfaces of the jaws and jaw slots, the back and front of the scroll, pinions, etc. and the inside of the jaw slots.
Jon