In my olden days, I used to like long, complicated tables (like these) for weather generation. Now I am more in favour of quick-and-easy methods for pretty much everything when I'm GMing, including weather generation.
So what I did, I bought a second 'weather die'. And I believe the combination of the two 'weather dice' is now giving satisfactory results.
If you do not have such a pair of dice (they're about £1 each in most shops), you can use 2D6 but it is going to be less fun. Let us describe the faces, and let us number them 1 to 6, for those who do not have the weather dice.
5= Partly Cloudy.
The illustration above shows the Raining (4), Partly Cloudy (5), and Fair (6) faces of the 'weather die'.
Use Of The Weather Dice to Determine Weather
Now, depending on the season you will use 1 or 2 weather dice.
Use 1 weather die only. Read the results as rolled, except Snowstorm (2) is replaced with Monsoon.
Use 2 weather dice and choose the "nicer" result, e.g., if you roll Partly Cloudy (5) and Fair (6), then choose Fair (6).
If you roll two 2's, then replace Snowstorm (2) with Hailstorm.
Use 1 weather die only. Read the results as rolled, except Snowstorm (2) is replaced with Drizzle.
Use 2 weather dice and choose the "worse" result, e.g., if you roll Partly Cloudy (5) and Fair (6), then choose Partly Cloudy (5).