Currently in minecraft you technically have only 3.5 weather patterns: Clear, Rain/Snow, and Thunderstorm. Hot and cold biomes do not get rain or thunderstorms though and only cold biomes or high elevations of biomes get snow. I suggest that all biomes should have at least 3 weather patterns:
Clear, Moderate, and Severe. Since all biome already have a clear weather in the game and some biomes share a moderate weather pattern I will only go over the biomes that are missing weather patterns.
-Hot biomes- Eventhough in the real world even deserts can still get rain and thunderstorms on occasion, we'll look at the world of minecraft a little differently since changing the frequency of how often it rains or storms is a bit boring. Instead I would suggest an atmospheric change. In a moderate weather phase of a hot biome the wind picks up and blows loose dirt or sand, decreasing visibility much the same as it does in other biomes that it rains in. For a severe weather phase not only would visability decrease to blindness level but there is a chance of an entity not in some form of shelter to be lifted off the ground and dropped in a slightly different location due to high winds.
-Cold biomes- Snow falls in place of rain in a cold biome so we will look only at the severe weather. Like with hot biomes there would be a blindness effect but a winter storm would also come with the danger of hypothermia, so an entity outside would have a chance of getting an effect like wither that slowly takes away hearts as long as the entity is "exposed to the cold."
This would take care of the overworld biomes currently lacking weather patterns but what about other dimensions?
-The End- It is difficult to think of the end having any kind of weather but it could still have cycles of some kind. What if in a moderate cycle there is an endermite migration. A few edermites will spawn randomly without the need of an enderpearl. During a severe cycle it is an endermite swarm passing though. As such, endermen are more easily aggravated by you due to the annoyance of the extra endermites they have to kill. Looking at any part of an enderman will cause them to attack, not just their eyes.
-The Nether- Lets face it. If you've been playing minecraft for a while you'll agree that the nether has lost the fear element it use to have. As such I'm going to suggest that instead of having a moderate weather phase and a severe weather phase that it should be 2 different severe weather phases. Phase one: Fire Storm- In place of rain fire charges accompanied with lightning will light the surrounding netherack a blaze temporarily. Like melting snow the game will eventually remove any blaze created by the storm. This will only happen in areas that have over 30 air blocks above an entity. Phase two: The Ritual Sacrifice. During an event like phase one any pigmen struck by lightning will seek out any entity that is not their own kind in an attempt to kill it in hopes it will put an end to the storm. The aggro will last till the storm passes.