The liturgical dedication of Saturday to Our Lady appears to have originated with Blessed Alcuin of York (735-804), an educator, scholar, theologian, and official at the court of Charlemagne, who composed unique votive masses for the several days of the week; he assigned two formularies in honor of the BVM to Saturday.  The Sarum Missal has the following explanatory preface before the Votive Masses of the Blessed Virgin Mary.

The reasons assigned at the beginning of this Mass for the origin of the Saturday in commemoration of our Lady, are — 1st. That at Constantinople the veil before her image was drawn aside every Friday evening at Vespers, and replaced at the same hour the following night; 2nd. That when all the disciples forsook our Lord and fled, she only who had borne him without pain, and knew that He was God, remained; 3rdly. Because the Sabbath is a day of rest, and she is the door of Heaven; 4thly. Because the Feast of the Mother should follow that of the Son (i.e. Friday); 5thly. For that on the day our Lord rested from labour the Service should be more joyous.

– Christian Clay Columba Campbell