George,
I believe she would be Violet Grantham, not Crawley. Crawley is Matthew's family. The Dowager is Lord Grantham's mother. (And I, too, think she is precious.)
The family name of the Earl of Grantham is Crawley. Robert Crawley has the
title of "Earl of Grantham," but that is just his title. His
surname is Crawley. It would follow, then, that his mother's name would be Violet Crawley.
Scroll down a bit on this Wiki page for a list of the cast . . . .
Downton Abbey - Wikipedia, the free encyclopedia
Consider the following from the same Wiki article on DA: "The device that sets the drama in motion is the entail accompanying the (fictional) Earldom of Grantham (which endows both title and estate exclusively to heirs male) complicated by the dire financial state of the estate only saved when the earl – then the heir apparent – married an American heiress."
So we have Robert Crawley, occupying the fictional Earldom of Grantham as "Lord Grantham." But his NAME is Crawley, and, of course, so is his mother's name Crawley. The key to understanding all of this is to not confuse the Earldom of Grantham with anyone's actual name. There is no one in the show with the surname of "Grantham."