If weather came with effects on mood and energy that would be an interesting addition. If it was rainy plants could grow faster, but you'd become miserable and soggy, to escape weather you could go to a friend's home and hang out. Maybe there could be a weatherman skill that could help you predict what sort of weather was happening and later on see the weather happening all over the map.
Today I thought, "How nice it would be to be in a beanstorm..."
You know, beans flying everywhere, Glitchens running around with their mouths open to catch them.... "This Weddingday will be cloudy, with a 20% chance of beanstorms in the Bortola area." Of course, this may have had something to do with my need for easily accessible beans at the time, but maybe it suits this world to have unique weather (perhaps cherries or bubbles or fish falling from the sky...).
I agree that weather affecting game play would be fun, but I don't know how difficult it would be. I can imagine the butterflies being happy to "sing in the rain" and giving you double XP (action effect), pigs wallowing in the mud afterwards (purely aesthetic), heat wave melting your ice (item effect), etc. etc.