I think the title "Mother's Rosario" comes from the religion of Yuuki (or her mother' religion) the number of shots in her unique ability along with the gift she makes to Asuna. 

"Rosario" is the Italian/Latin (no Portughese) word used for a form of prayer in the Roman Catholic Church. In its short form, it consist in a sequence of prayers (one Apostles' Creed, one Lord's Prayer, three Hail Mary, one Glory Be) followed by ten Hail Mary that must be recited in a loop at least five times.  

A necklace, called "Rosario" itself,  wich is componed by a string of knots or beads is used in a hand to maintain sequence and count of prayers.   The necklace name is derived from latin "Rosarium"  wich stands for "crown of roses" and, among the Catholics, there is the tradition of gifting these necklaces.

PS: I'm Italian and I'm Catholic.