What does this mean? It is on the bottom of a strange coffee mug
The characters are "悠游三国喜神方". Proof link here.

三国 (Three Kingdoms) refers to the story of Romance of the Three Kingdoms.

悠游三国 (leisurely visit the Three Kingdoms) is an online game.

喜神方 (happiness god's way) is a concept from old Chinese's belief in luck. When the almanac says one shouldn't travel, there are methods to find "happiness god's way" following which one may travel without getting the ill luck. I believe in your phrase, 喜神方 refers to that method (maybe it's a ceremony in the game).

Put them together, 悠游三国喜神方 means 'the method to understand happiness god's way (so one may travel without getting bad luck) in the online game You You Three Kingdoms'.