Devotion to Mary under the title of Our Lady of the Snows is one of the oldest devotions to Mary. It has direct ties to the legend about a marvelous snowfall in Rome in 352 A.D. Mary had indicated in a dream to a wealthy, childless Roman couple that she wanted a church built in her honor and the site for this church would be covered with snow. On a hot, sultry morning on August 5, Esquiline Hill was covered with snow. All Rome proclaimed the summer snows a miracle, and a church to honor Mary was built on the hill in 358 A.D. Restored and refurbished many times, this church, now the magnificent Basilica of St. Mary Major, still stands today as the seat of devotion to Our Lady of the Snows in the Catholic Church. However, Our Lady of the Snows is honored here, not so much because of the legend, but because of her special role in the Church that is by its very nature, missionary.
-Excerpt from snows.org