My experience has been that very few games incorporate weather into their depictions of overland travel. This is usually because it's too much hassle to resolve what the weather is and how it impacts the PCs, without necessarily providing very interesting results as the pay-off. I think that by creating simpler systems that produce results more quickly and don't bog down play, we can get weather determination back into game-play as something fun that helps flesh out the feel of the world.
I've written several procedures for determining the weather over the years with varying levels of complexity and randomness. I've also experimented with making it a task that players perform (since it's information about the world that's immediately apparent to them). After all of that experimenting, I've found that the best combination of simplicity and granularity is to simply use reaction rolls for weather.
What we really want to know is "Does the weather aid or hinder the PCs?" and maybe get a prompt for depicting that in the fiction. And because weather is almost always a minor supplement to whatever else is going on, we want to get it done quickly, and not use a flat probability distribution that's going to overpopulate the game with inconvenient weather (unless you're playing a game set in England, of course). That would have to vary seasonally and regionally anyhow to properly convey verisimilitude, which adds a lot of background work on to the ref for little pay-off.
The 2d6 bell curve with five levels of results inspired by Moldvay's B/X version of D&D gives you a pretty simple way to adjudicate it and a clear sense of what's going on. NB: I'm basing my examples below on Courtney Campbell's inversion of the table, where ascending results are good instead of bad. I actually use a slightly simpler (three levels instead of five) in my home games, but most readers here will be more familiar with Moldvay's B/X than that, I think.
A result of "2" for "Hostile" means the weather is a genuine risk to the PCs, and they need to stop whatever they're doing (marching overland mostly) to deal with it. I usually include the risk of some damage, either to them or their gear or both, if they don't. "3-5 Unfriendly" means a slight hindrance - mostly slower travel, or the need to expend some resource (rations, water, etc.). Neutral results (6-8, and the most common kind) don't have any effect, while "Friendly" results (9-11) will help you travel slightly more quickly or expend fewer resources, and "12 Helpful" speeds you along or lets you avoid perils.
I like the emphasis on functionality here rather than specific results because it gives you the space to adapt the results to the fiction - fog can be a help or a hindrance, depending on the situation. Winter and summer have radically different kinds of helpful or desired weather.
If you're stuck for ideas, you can also roll 1d4 to tell you what the biggest change is:
Helpful precipitation (a roll of 12 on 2d6 and 4 on 1d4) might be a light rain that sucks the humidity out of the air in summer, or it might be a crisp snowfall that lets you ski across it easily in winter. "Hostile visibility" could be pea-soup fog, snow blindness, or heat waves washing out the horizon and creating mirages.
I roll about three times per day of travel - twice for the two four hour chunks the PCs are marching around, and once for the overnight weather, but alter the frequency to your own taste. I also have the PCs roll it instead of doing it myself, since the quality of the weather is something that should be immediately apparent to them anyhow. And with the 1d4 prompt, they can actually come up with even better suggestions for "helpful" or "friendly" results than I might, since they know what gear, knowledge and talents they have better than I do.
I recommend trying it out if you haven't and seeing how using reaction rolls like this allows weather to be incorporated with minimum fuss.