My suggestions for weather:
Light Rain - light rain would be a common occurrence happening around once or twice a year in game time, and lasting for a few weeks to a month. When light rain occurs the popularity of rides would go down unless they were covered. What I mean by that, is if you've built a ride that's underground, or has a roof over it(like a dark ride), it's popularity would remain unaffected.
Heavy Rain/Sandstorms/Snow - heavy rain (or sandstorms for the desert biome, snow for winter biomes) would be uncommon, only occurring about once every 1-2 years, and lasting about a month in game time. When this occurs the popularity of rides goes down even more than when light rain occurs, and guests will now actively seek covered areas. Covered rides would also now become more popular than they would otherwise.
Thunderstorms/Blizzards - thunderstorms (or blizzards for snow biomes) would be a rare occurrence, maybe happening only once every 2-3 years in game time, and lasting roughly 1-2 months. When thunderstorms occur, rides over a specified height would be forced to close down. But unlike ride breakdowns, guests would remain in line and would not be issued a refund. No new guests would allowed to buy tickets, and ride operations would simply just cease until the storm passed. Flat rides and tracked rides that don't exceed the height limit could still operate, but their popularity would drop significantly. In turn, rides that are covered/indoors would receive a significant boost in popularity.
Campaign - maps could be interesting, having some where rain and thunderstorms occur more frequently, increasing the difficulty.
Challenge an sandbox - each biome would have its own unique weather, including frequency and type. Tropical biomes might have have more frequent storms than an alpine one. Deserts would have sandstorms and snow biomes would have blizzards. These would both probably need to be much more infrequent to offer variety but also maintain realism
Options - Just like with all new gameplay elements, weather should have an option to turn off for those who don't want it.
Overall I think this would add a pretty interesting game mechanic. The frequency of the storms occurring would need to be spread out so you don't run into a year where it's constantly storming or raining to some extent.