From the Anglican Breviary...
"THIS feast is in commemoration of the first church to be dedicated in Rome under the invocation of our Lady, and the third of those Christian temples in the City known as Patriarchal Basilicas. The origin of this building, according to an old story, popular in ancient times, was as followeth. In the middle of the fourth century, during the pontificate of Pope Liberius, there lived at Rome a certain nobleman named John, and a noble lady his wife, who had no children to whom to leave their substance, and who vowed that they would make the holy Virgin Mother of God their heiress. And earnestly they besought her in some way to make known to them upon what godly work she would have their money spent. And thereupon (so saith the story) the blessed Virgin graciously listened to the heart-felt earnestness of their prayers, and by a wondrous sign assured them of her will.
"ON the fifth day of August, which is the time when the heat of summer waxeth greatest in Rome, a part of the Esquiline Hill was covered at night with snow. And some scholars think that such a strange and unseasonable fall of snow did take place, and so gave rise to the old tale, which goeth on to say on this same night the Mother of God appeared in a dream to John and his wife separately, and told them that on that spot, which in the morning they could see clad in snow, they should build a church, to be dedicated under the name of the Virgin Mary, for that this was the way in which she chose that they should make her their heiress. Then John went and told it to Pope Liberius, who declared that he also had been visited by a like dream.
"THEREFORE Pope Liberius went in a solemn procession of clergy and people to the snow-clad hill, and traced upon that spot the plan of the church, which same was afterwards built with the money of John and his wife. And later it was rebuilt by Saint Pope Sixtus III. At the beginning it was called by divers names, sometimes the Liberian Basilica, sometimes the Church of Saint Mary-at-the-Manger (because of the presence there of a relick revered as the Manger in which our infant Lord lay), and so on. Howbeit, since there are in Rome many churches called after the holy Virgin Mary, and this church, both in age and dignity, doth excel them all, it is commonly called St. Mary Major. And the memory of the dedication there of is kept every year by this feast-day that taketh name from the strange fall of snow which is said to have taken place on this day."