This year has been an oddly mild winter for my hometown, which happens every handful of years. For example, last week, we had one day that was 30 degrees (F) warmer than the average. I comfortably wore a short sleeve shirt and jeans while walking the dogs.
The next day, it was an average day. I wore a coat and was considering breaking out my ear muffs and gloves.
Most of the residents shrugged it off. It's not uncommon for us to have relatively extreme swings in the weather. We even make jokes about it--"You know you're from ---- if you have to run the heater and the A/C in the same day."
On the other hand, the average highs for San Diego, California vary about 10 degrees Fahrenheit all year.
Let's just say I don't live in California. (That leaves you 49 more states to choose from, if you're trying to track down my secret identity ;) )
Even if you don't decide to unleash a natural disaster on your characters, mundane weather can cause quite the stir and provide an opportunity for conflict amongst your cast.
If you have a character from a warm climate who has to travel to a cold climate, perhaps they can't find the clothing they feel they need to stay warm. What if they don't know how to use an ice scraper to scrape off their getaway vehicle? What if they don't know how to drive on snow? You can always have them be confused by the buckets of salt/ice melt that are kept by the doors if you need some comic relief.
Conversely, have your cold-weather character confused by why air-conditioning is necessary, or how hot the insides of cars get in the summer sun, or the fact that you can theoretically fry an egg on asphalt--and that there's a town with a festival devoted to doing so.
How do your characters handle sunburn? Windburn? Frostbite? Are these concepts that they're familiar with?
How does the climate or weather affect the style of housing your characters live in? For example, historic homes in the American South typically have large breezeways so that there can be more airflow on stifling summer days. Areas that receive high amounts of snowfall are likely to have buildings with steeply pitched roofs that allow the snow to slide off.
How does the climate affect what foods are available to your characters? How about what foods they want? (Ex. a person in a hot climate may be less likely to spend time developing "heavy" meals or stews, but focus on lighter fare instead).
What's something quirky about the weather where you live?