We're starting up a dark ages fantasy game, using some homebrew rules similar to D&D.
As the GM, how can I make weather an important part of the game?
To be clear, I'm not trying to just toss in random weather details that the players will ignore. If I say at the start of each game day that it's "partly cloudy" or "lightly raining", they'll all just ignore it, because it doesn't matter to them. What can I do to make weather really make a difference?
Weather affects the overall atmosphere in the game. To make it relevant, work it into the descriptions of places the PCs visit, people they meet, and challenges they face. Here are just some examples:
Rain
Snow
Drought
...you get the idea. Weather can be used as a random encounter modifier, dice modifier, atmospheric flavor bit, or even a plot hook.
I would add mechanic bonuses as others have mentioned but I would also makes changes to the other things that are in your control as a GM. The three that spring to mind are NPC, Monsters and the Environment. Weather could be used to affect them like so:
NPCs react differently or break promises.
Monsters act differently
Weather affects the environment.
I think these narrative hooks are more engrossing than the mechanical ones. While mechanic updates make the choices players make more involved, narrative effects add depth to your world.
Probably the best way to make anything relevant in a game is to make it consistent. Even small bonuses can make a difference, and if you are careful to point them out to the players, they'll catch on soon enough.
To that end, you can have a chart drawn up of what the weather will be like for the next year or so in game time, and keep track of what day it is when adventures happen. And it doesn't have to be a bonus to rolls, either. Things can cost more money during droughts, or passages will be cut off during floods. Or things will simply take more time. Raining? It'll take an extra day/week to get where you're going.
The prerequisite to this is that you have to have time mean things in game in the first place. You have to get the antidote to the palace by midnight on the third day to prevent the king from dying of the Wasting Sickness; the enemy horde is headed for the City Gates and you have to beat them there in order to warn the guard.
Alternately, have small bonuses or penalties based on how wet, cold, hot, or windy it is. Maybe the PCs get attacked on a slippery pass on the side of the mountain by flying beasts, and have -2 penalties to all of their attacks and DEX checks.
(I have a lot of assumptions about the game you're running, coming from an Old School background; specifically, the PCs can fail, and can die. Maybe going around the mountains will be the safer, but slower choice. In any case, weather affects resources like time, food, and money, and to go by this route, you have to keep track, at least approximately, of those things.)
Make yourself a list of reasonable effects for exposure to different weather conditions. Then, when by whatever method you select the weather for a given span, you can apply the effects.
So something like:
If you want random weather, I would build a set of tables for my current environment / season on something like 3d6 with the most common weather conditions in the center and less common conditions near the extremes. That way, you'll tend to get the expected weather, but you could always get something harsher / nicer than expected. So...general climate goes in 9-11.
Another aspect that might be important is the effect weather has on attitudes and interpersonal dynamics. Think about all of the cultural artifacts at the macro level around the change of seasons, particularly in societies that are connected to the land:
Winter snow and rain is dreary, people get the blahs, they try to stay inside. Getting them to do anything outside can be difficult. Spring, with its mix of rain and sun, is the time of rebirth and renewal. People are more relaxed and playful. There are celebrations around fertility, the planting of crops, and so on. Summer is a mixture of the labor of tilling fields and the enjoyment of the warm, sunny skies. Fall is a time of preparation, feasting, and reflection, as the heat recedes and the rains start up again.
On a micro level, when you're out in the elements, as adventurers are, these effects are magnified. Trudging on and on through a cold, driving hailstorm is not just physically difficult; it is a test of will. You could hunker down and make a shelter. You could turn back and go find a warm tavern. But here you are, continuing on your quest. It may not be a good idea to force player characters to make willpower rolls in order to press on, but imagine if an NPC failed his roll and decided to throw in the towel. "Is finding The Eye of Schoban worth losing my toes over?! I think not!"
The easiest sorts of weather effects to portray are extremes, in particular of hot and cold. When you are all bundled up and still have no heat, there's a slow, gnawing feeling that sets in, like you're ebbing away. It's insidious. All you want to do is preserve whatever last ember of heat remains in your body: "Walk three feet to ring the alarm bell? Uh, no. I'll curl up in a ball here, thank you very much." Again, how you want to introduce the willpower element is up to you, but there should be some challenge put in front of the players, even if it's purely roleplaying, such as making them explain to you how they're going to keep themselves focused on the task at hand, rather than on the cold.
Heat is different, in that it makes you want to escape yourself. When you're broiling, there's an intense desire to become non-corporeal, to escape your body, to drift off to someplace where the heat can't get you. If cold is the absence of a form of sustenance, heat is a brutal, in your face assault. Where cold makes you not care, heat makes you agitated and angry: "Derbold, if you whistle that song one more time, I'm going to rip your throat out." Where cold will lead to dissipation and a slow collective wasting away, heat will lead to violence if there is any excuse for it.
Also don't forget that how well or poorly the PCs prepared for whatever weather they're in can make a huge difference. Thoughts of the first Soviet invasion of Finland and the German invasion of Russia come to mind.
You can have the PCs' opponents make use of the weather. Particularly with weather magic, but also taking advantage of weather conditions to put one over on the PCs.
So, first, weather magic. Going near the tower of Boron the Weathermage in cloudy weather is foolish; he'll use his magic to call down lightning. Doing so in a storm is actively suicidal, because he gets a contact high from it, and consequently acts like he's drunk. Clear days are really the only way to get there.
On the other hand, approaching the domain of Gos the Artificer on a clear day is madness; he uses magical mirrors and magnifying glasses to focus the sun's rays into killer beams of heat and light. And you don't want to know what he can do with moonlight.
Make the weather known, and then, if the PCs don't take account of it, apply consequences. Marching in your armour on a hot day? Fatigue penalties. Didn't bother with the waterproof tents in the rain? Spellbook soaked, everyone has sniffles, which impose penalties on fighting and social roles.
Dedicated enemies can move past PC scouts in bad weather. Storms hold ships in port. Floods destroy bridges, block travel routes, and may even cause enough damage to make all the peasants in that area flee elsewhere, giving streams of refugees on the roads and overcrowding in nearby towns. And where there's overcrowding, crime and disease rise.
Add minor effects of the weather on magic? Rain = +1 to lighting based spells, sunny sky +1 to fire? +1 on 5d6 might not be much, but knowing players they want every point they can get.
Get an old copy of the Wilderness Survival Guide, it was designed for AD&D 2nd ed, but most of it is system agnostic. You'll have to modify any combat or damage mechanics to suit your game, but most of it will still be relevant.
Lots of great answers here, but I want to give it a shot with my own perspective. You're really asking how to make in-game weather matter to the players.
I'd touch on all aspects of the game, including social contract, setting, characters, situation, and system.
Social Contract
By which I mean the agreement of all participants to play this game, at this time, in this way. This is the fundamental layer of all play. As such, it's the most important and the first place I'd focus.
Tell your players you want to make weather an important part of the game. Explain why. Get them to buy into this idea. If they're not behind it, nothing else you do will work.
Here's the thing: generally players don't care about weather. They want to play larger-than-life heroes and do heroic things. Getting rained on or getting a sunburn ruins the magic. If weather is going to play a part in your fantasy game, you have to make it cool.
Setting
Other folks have offered ways to bring weather into your setting. Your game is fantasy dark ages, so you don't have to stick to boring, real-world weather patterns. Go big. Thunderstorms should split castles in half. The Summer sun should burn people's skin in minutes if they're unprotected. Decide that all major weather is caused by actual elementals. The characters can then go fight the weather.
Characters
Tie every character to the weather. Everyone gets a weather sign, like the Zodiac but more chaotic. If you were born under the rain, then you get some kind of perk when it rains. Now every player is asking you what the weather is like, because they want to press the button on their character sheet. Replace a bunch of character abilities (especially the magical ones) with weather-specific versions. Eschew magic missile; embrace lung rain (which does the same thing mechanically).
Situation
Situation is where character and setting meet, and system carries it forward in time. Build world events around the weather. For epic fantasy, consider a global warming plot, only faster acting. Every month, more weather elementals enter this world and they're wreaking havoc: dust storms, tornadoes, tsunamis, volcanoes, droughts, floods, and so on.
Too gonzo for your game? Make sure the in-game weather is a strategy element that the characters (and their enemies) can use to their advantage. Lots of real-world battles were won or lost because of rain or fog. If the characters have some means to actually control the weather, then they can use it more, of course.
System
Show the players a big red button and they will push it. If the button gives them some advantage, they'll figure out how to push it as often as possible. Rules that the players can use to take advantage of in-game weather are that red button.
A lot of the other answers talk about stuff for the GM. That's great, but it won't hook the players. They need to get their hands on the new rules for you to grab them. Figure out what new rules every player gets to interact with the weather systems in your game.
I'd like to add some more real-world weather effects, that could affect the party too. Things get interesting, when you are near water melting/freezing point (0 degrees C), i.e. in the mountains:
All these "disasters", though small, can be really irritating, and even cause bad accidents when ignored. But the effects can be cancelled or softened by making right decisions, such as choosing when to wake up, where to set you bivouac and if to go futher or stop.
There are alot of different importants. In regards to one type of important, make various penalties for weather effects BUT also make actions the players can do to mitigate the penalty and they can't just do them when the weather happens, they have to keep doing them. So maybe the cold makes your sword stick in it's sheath, making the first attack -3 to hit. But if the players, every so often as part of their roleplay, say they oil or move their sword in their sheath, it becomes only -1.
