Weather table examples wanted

Not a direct answer to your question, but:
There was an old turn-based computer game called Romance of the Three Kingdoms set in ancient china. During battles you could start fires which spread according to wind direction. If your game has a grided board, you could use fire tiles and an six or eight-sided teetotum (dice-like spinning top) for wind direction (depending on whether you use a hex or square grid). You could expand this further for other weather types - so if it is raining, then fire won't spread, but on hot dry days, the fire spreads faster. You could replace the teetotum with dice or a four, six or eight-card "wind deck" (maybe look at the wind tiles in mahjong for inspiration)

Also if it were raining, movement across certain types of terrain (eg. swamp or river crossing) might incur a penalty of some kind for most units. Ditto frost or snow (there were several accounts of famous knights and at least one king in medieval Europe who died as a result of slipping on ice while fighting or on the way to a battle).
Perhaps rivers and lakes could freeze over in winter, allowing quicker crossings while the "rivers freeze" card is in play, with a random risk of damage to heavier units.

If there is a naval element to the game, wind should affect the movement of ships (my teetotum idea was originally used in a pirate-themed game, where ships moved slower upwind and faster downwind).

Fog (and smoke, if you include fires) would be nice to include, but I'm not sure how one could do this simply. Maybe ranged units like archers would be penalised in their attack, or there would be a small random chance of attacks hurting nearby friendly units "by accident", or you could just make the outcome of combat more random than normal to simulate the confusion caused by a lack of visibility.

I like some of the ideas already mentioned. It's simple, and not particularly interesting, to simply make weather provide numerical bonuses or penalties, such as rain provides a -1 penalty to attack rolls.

However, the weather effects that modify the board state will create more interesting decisions for players, forcing to them to rethink new strategies. For example, rivers freezing over in winter effectively change the game board by allowing movement in a direction previously not allowed (or severely hindered). Vegetation during the spring and summer months might allow gathering of food and herbs, which can be used for ointments. Maybe during the fall, it's difficult for heroes to be stealthy due to all the foliage on the ground.

Anyways, there are some great opportunities to make weather more interesting than simple, numerical penalties.

Good luck!