Mary, her mother lived to be +100.
No; her mother was Elizabeth Bowes-Lyon Windsor, who lived to the age of 101. Bowes-Lyon was the daughter of Scottish nobles. From the time that the present Queen Elizabeth II ascended to the throne in 1952 at age 25, her mother was then known as "Queen Elizabeth the Queen Mother."
The current Queen Elizabeth II's paternal grandmother was Mary of Teck, who lived to the age of 85.
Elizabeth II is not "the second" because her mother was named Elizabeth, but because there was a previous Queen Elizabeth in the British line of succession. Elizabeth Tudor, the daughter of Henry XIII, reigned from 1558 to 1603.