A change in weather making a person sick is just an old wives tale and holds no truth to it.
Weather cannot make you sick and the cold weather doesn't bring on a cold.
Colds and the flu are brought on and caused by viruses that people get inside there body.
During cold weather people are usually inside around other people more and then if that person has the cold or flu virus then it's much easier for them to catch a cold or flu.
I work in an office and during last summer I caught a cold from a coworker who sits next to me.
The coworker had a cold and the next day I came down with a cold so it's not just winter when you can catch a cold you can catch a cold or even the flu anytime of the year.