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.
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.
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.
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 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.
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.