The descriptions of nature or weather in the story reflect Granny Weatherall's state of mind. The morning of her death her bed felt "pleasant as a hammock in a light wind". However, as she nears her death, "a fog rose over the valley, she saw it marching across the creek swallowing the trees and moving up the hill like an army of ghosts. Soon it would be at the near edge of the orchard." The fog represents her loss of alertness as she nears death, which is the orchard. The fog swallows the trees representing the fogginess that is covering her consciousness; the fog moves like ghosts; this image represents her impending death.
The descriptions of nature/weather also mirrors the day of her supreme disappointment. Before she is jilted, there is a pleasant description of nature which represents her hope for the future "such a fresh breeze blowing and such a green day with no threats in it." However, after her fiance jilts her, the light of day is diminished as a "whirl of dark smoke rose and covered it, crept up and over into the bright field."
When Granny is on her final journey toward death, she rode in the cart and looked "down the road where the trees leaned over and bowed to each other and a thousand birds were singing a Mass." Here she is entering her final state of eternal spirituality
Weather is always changeable. It is unpredictable, and just blows through lives. Think of all the things she remembers about her life; she's been through a lot of weather.
But there's a more direct meaning as well. To "weather" is to get through—to stand up to something, like a barn that lasts through a storm. Granny has weathered a lot. In fact, she's weathered all. She's tough and has been through lots, and that's exactly what her name tells us.