Weather simulation can be as simple as a markov chain or as complicated as a simulation of the physical processes that lead to weather.
PCG Wiki References
- Dwarf Fortress uses weather simulation in its generation of maps and simulation of areas.
Technical Demonstration: Far Cry 2 - Video of procedural sky, dynamic weather, and fire propagation in action.
Atmospheric model - Wikipedia article on numerical weather prediction models.
Climate model - Wikipedia article on climate simulations.