First idea - Raining and sunshine (in the overworld) have respective weather uses and benefits in battle. Speed rise for Pokemon with Swift Swim, Chlorophyll, etc. abilities. Also, Rain Dance, Sunny Day, etc. can change the overworld weather temporarily. This feature would be useable in the same way other field moves could be. Dig, forage, etc. Maybe the weather effects could last a few minutes, like a normal spell in Vanilla Minecraft. This could add an interesting element of surprise and uniqueness to certain biomes. For example, using hail in a Jungle biome to cover everything with a temporary layer of snow. Using Sandstorm in a Forest biome to cause random blocks of sand to fall from the sky and then litter the landscape. (I understand this might make for unwanted sand blocks. Perhaps if not mined, the sand would eventually disappear.) Similarly, adding a Sandstorm feature in the desert sometimes could reduce visibility and add other interesting twists to exploring this biome. (Effect on Pokemon spawning described later.) But adding the Go-go goggles would serve as a way to "see through" the storm and continue as normal. This is exactly like the night vision and underwater vision associated with those two conditions.
Second idea - During thunderstorms, lightning is attracted to/strikes near Pokemon with the lightningrod ability.
Third idea - Legendary Pokemon cause weather to change in the areas they spawn. Raining above water in an Deep Ocean biome for Kyogre or perpetual daytime - motionless sun for Desert Hills for Groudon. Leaving the biome returns the game to normal.
Fourth idea - certain Pokemon spawn more readily under certain weather conditions. For example, during thunderstorms, electric and steel Pokemon are more likely to spawn in a respective biome. Or during a sandstorm, Cacnea or Gligar might be more likely to spawn because of the sand veil ability.
Fifth idea - Certain pokemon ONLY spawn under certain weather conditions. For example, Snover would spawn ONLY if it were snowing/hailing. (Or NOT under certain weather conditions.) For example, Charmander would NOT spawn while it's raining, because it would extinguish his/her tail.
Okay, that concludes all my weather-related thoughts for right now. I'll add more as I think of them. To anyone, please let me know your thoughts on these suggestions.