Now with rules! I made these rules up, based on DMG rules and Regional effects, but they're probably no better than the rules you would make up yourself. It's more like an example for novice DMs, or for those DMs who find it easier to look up rules than make up rules.
Princes of the Apocalypse
Weather results in the Dessarin Valley
1d8:
1 Pleasant
Tomorrow, if you roll a 2-6, there is chance you will treat it as a 7 (Extreme Weather) instead. The chance is 25% per Pleasant day in a row.
2 Pleasant, but several minor tremors throughout the day
The tremors are not severe enough to damage creatures or structures.
Each round of combat, there is a 10% chance of a tremor. All creatures standing on the ground must make a DC 10 Dexterity saving throw or fall prone. Creatures with more or fewer than 2 legs have advantage on the saving throw.
While traveling overland, PCs have a 10% chance of encountering a landslide in hills or badlands, or a 20% chance in the mountains. Spotting the landslide ahead of time requires a Passive Perception of 15 or success on a DC 15 Wisdom (Survival) check. Each character must make a DC 10 Dexterity saving throw, taking 6d6 bludgeoning damage on a failure or half as much on a success. If the landslide was spotted ahead of time, characters have advantage on their saving throw and take no damage on a success.
3 Pouring rain
Overland travel speeds are halved, and everything is lightly obscured.
The DC of all tracking attempts increases by 10.
Characters near a river, lake, or swamp have a chance of encountering floodwaters equal to 20% per day of rain. Noticing an impending flood requires a DC 15 Wisdom (Perception) check, and allows characters to make a Strength (Athletics) or Dexterity (Acrobatics) check to flee to higher ground before the floodwaters hit (the DC is between 10 and 20, based on how close the group is to higher ground). Characters swept up by the flood must make a series of DC 15 Strength (Athletics) checks to keep afloat. After 1 failure, the character takes a level of exhaustion. After 3 failures, which need not be consecutive, the character begins to drown. After 3 successes, which need not be consecutive, the character swims out of the floodwater, unless they find some other way to escape the flood sooner (such as being helped out by an ally).
4 Sunny, hot, and dry (or humid, if it rained yesterday)
Every 4 hours that a character travels overland, they must succeed at a DC 10 Constitution saving throw or suffer a level of exhaustion. Characters in heavy clothing or medium or heavy armor have disadvantage on the saving throw, and characters resistant or immune to fire automatically succeed.
5 Cold and windy
Wisdom (Perception) checks based on hearing are at disadvantage.
Overland flight speed is halved.
Characters without warm protective clothing or resistance or immunity to cold must make a DC 10 Constitution saving throw every 4 hours they are exposed to the wind. On a failure, they take a level of exhaustion. Characters engaged in vigorous activity, such as overland travel, have advantage on the saving throw.
Each round of combat, there is a 50% chance of strong winds, or 25% if the characters are in a protected environment such as a box canyon or dense forest. During a round of strong winds, ranged weapon attacks are at disadvantage, and flying creatures must land at the end of their turn or fall.
6 Overcast, with rumbles of thunder and strikes of lightning
Magnetic disturbances cause compasses to malfunction.
When a creature takes lightning damage, the DM randomly selects 1 creature within 15 feet (including the creature that took the initial damage). The selected creature takes a further 1d10 lightning damage, which doesn't trigger this effect.
7 Extreme weather, roll 1d4:
1 Snowstorm (or Hailstorm, in the summer time)
The storm lasts 2d4 hours. Predicting its onset requires a DC 10 Wisdom (Survival) check.
Snowstorm: Use the rules for Extreme Cold (DMG p.110). Each hour produces 6" of snowfall; ground with 1 foot or more of snow is difficult terrain. Snow melts at a rate of 1 foot per day, or 4 feet per day when it is hot out.
Hailstorm: Characters out in the open are pelted by hailstones, and each round must succeed at a DC 10 Dexterity saving throw or suffer 1d4 bludgeoning damage. When caught out in the open, characters can improvise a shelter with 1d4 rounds and a successful DC 10 Wisdom (Survival) check. Waiting out the storm in such a shelter still requires a DC 15 Constitution saving throw; on a failure, the character suffers a level of exhaustion.
2 Tornado
PCs have an 80% chance of encountering a tornado in plains or badlands, a 20% chance in the mountains or on a lake or river, and a 50% chance elsewhere. Predicting the tornado's arrival requires success on a DC 15 Wisdom (Survival) check and gives the characters 10 minutes to find shelter. Spotting it requires a Passive Perception of 15 or success on a DC 10 Wisdom (Perception) check and gives the PCs one minute to find shelter. Shelter can be located with a DC 10 Wisdom (Perception or Survival) check, with disadvantage if you only have 1 minute to get there; or the PCs may already be near obvious shelter, such as a stone building. PCs under shelter when the tornado strikes must succeed on a DC 10 Constitution saving throw or suffer a level of exhaustion. Characters out in the open take a level of exhaustion and are pelted by debris for 2d6 bludgeoning damage. They must then make a DC 20 Dexterity saving throw to evade the tornado. On a failure, they must make a DC 15 Strength saving throw. On a failure, the character takes 3d6 bludgeoning damage and is flung 3d6x10 feet up into the air. On a success, the character half damage and is not flung.
3 Earthquake
The earthquake is severe enough to destroy buildings and cause avalanches. Characters outdoors on relatively level terrain are safe from these effects. Otherwise, characters must make a DC 15 Dexterity saving throw. On a failure, characters in stone buildings or mountainous terrain take 10d6 bludgeoning damage, and characters in wooden buildings or hilly terrain take 6d6 bludgeoning damage. On a success, characters take half damage.
There is a 20% chance of a fissure opening beneath the characters (whether or not they were caught in an avalanche or building collapse). Anyone standing in the area of the fissure must succeed on a DC 10 Dexterity saving throw or fall 2d6x10 feet into it.
4 Wildfire
Use the rules for Extreme Heat (DMG p.110).
PCs have an 80% chance of encountering a wildfire in grassland or forest, a 20% chance in swampland or near a river or lake, and a 50% chance elsewhere. Spotting it more requires a Passive Perception of 15 or succeeding on a DC 10 Wisdom (Perception) check. The wildfire moves 50 feet per round and is 2d4x100 feet wide. Characters who spotted it ahead of time have 3d4 rounds to escape; otherwise the wildfire appears only 1d4x50 feet away. A character in the area of the wildfire must make a DC 15 Constitution saving throw, taking 2d10 fire damage and a level of exhaustion on a failure, or half damage and no exhaustion on a success.
8 Same as yesterday (for Extreme weather, re-roll 1d4)