Having fished up every fish in the game and meticulously watched the weather for many a fun-filled hour, I can tell you with 100% certainty that weather does rotate on an 8-hour cycle - 12am to 8am to 4pm to 12am. Now, what weather you get when it changes is completely random within the possible weather patterns for that area, and sometimes it just chooses the same weather it was before, so you don't see any change.