On the Nones of August, usually the hottest time of the year in Borne, a part of the Esquiline hill was covered with snow during the night. That same night the Mother of God appeared in a dream to John and his wife separately, and told them to build a church on the spot they should find covered with snow, and to dedicate it tot he Virgin Mary; for it was in this manner that she wished to become their heiress. John related this to Pope Liberius, who said he had dreamt the same thing.
He went therefore, with a solemn procession of priests and people to the snow-clad hill, and chose the site of a church, which was built with the money of John and his wife. It was afterwards rebuilt by Sixtus III. At first it was called by different names, the Liberian basilica, St. Mary at the Crib. But, since there are many churches in Rome dedicated to the Blessed Virgin Mary, and as this one surpasses all other basilicas in dignity and by its miraculous beginning, it is distinguished from them also by its title of St. Mary Major. On account of the miraculous fall of snow, the anniversary of the dedication is celebrated by a yearly solemnity.
Source: The Liturgical Year, Vol. XIII. 1868