So where does the name Mary come from:
In a sense, Mary needs both of those qualities for the work to which she is called. She is the beloved one, cherished of God, who is selected to carry and give birth to the Messiah. When the angel Gabriel first appears to her to tell her about her role in the Christmas story, he calls her "favored one."
On the other hand, that rebellious streak -- the ability to follow her inner voice when others might try to convince her to behave differently -- will come in handy throughout her pregnancy.
Her courage allows her to travel to Elizabeth and Zechariah's when the angel tells her to go. It sustains her through her and Joseph's journey to Bethlehem and Jesus' birth out in a stable. Most importantly, her rebellious nature gives her the ability to say yes to being used by God in this way when she knows the consequences could be dire.
We need to be more like Mary, rebelliously courageous because we know we are loved by God.