Weather

Using the DMG, random weather can be generated. I have fine-tuned the generation a little to take zones such as mountains, valleys, and deserts which have different weather scales, as well as seasons, into account. But the application of weather in terms of the mechanics involved in RAW is still lacklustre. This alternate system of weather mechanics is adapted from this post. The changes I made were mostly to allow for the greater impact of exhaustion in a game where long rests are exceedingly rare.

Random Weather Generator

Note: The word usage here for temperature does not necessarily align with the temperature labels in the table below. The generated weather should be adjusted and clarified to fit the table below when desired.

Mechanics

This may seem like a lot of information. The system is simpler than it looks though. The first table is the important one. It breaks weather into temperature brackets, and sets a penalty for being in that bracket. The rest of the tables describe different weather- and character-related things which affect your experience of the temperature, mostly by moving you up or down a bracket.
I didn't come up with this system. Credit goes to this post.

Hot and Cold brackets have a DC and damage time scale. Players in that bracket must make a daily Constitution saving throw against the given DC or get 1 level of exhaustion. Every given time unit, they take 1d4 heat/cold damage.

* Creatures must drink a double portion of water or make the save with disadvantage.

Modifiers

The following tables describe different modifiers to the above table. To move a step "up" means towards hotter brackets, and "down" means towards colder brackets.

* Determined by the DM.

Example

It is winter in the Misty Mountains, and the weather is extremely cold with light snowfall and moderate winds. Bob wants to cross the mountains, but he knows it's going to be cold so he prepares. He is wearing cold weather clothing, and he is wearing his heavy armor for protection. Bob grew up in the Misty Mountains and is accustomed to cold, so the DM rules that he takes half damage from cold. Feeling cold, Bob stops for lunch in a cave and makes a fire. He drinks hot soup, takes a short rest, and then continues.

The temperature is Extremely Cold, the light snowfall moves it down a step, and the moderate winds move it down another step. Bob's clothing moves him up two steps and his armor moves him up one step. The fire at lunch is a warming apparatus and moves him up another step. This places Bob in the Cold bracket, and he will make a Constitution saving throw against a DC 12, and take 1d8 cold damage every hour, resulting in a total of 8d8 damage over his 8-hour day's march! Fortunately, he takes half damage from the cold, and his hot soup means that he will make the Constitution saving throw with advantage.

The next day, the winds continue, but the sun comes out. Without moving down a step from snowfall, Bob ends up in the Cool bracket, meaning that, although crossing the Misty Mountains in winter is still not very bright, Bob will make the days march with no negative effects.

DMG Tables

The random weather tables from the DMG: