From the experts:
This attractive Scottish surname choice, has a certain amount of glamour thanks to feisty British-born red-haired forties Academy Award winner Greer Garson, who was born Eileen Evelyn Greer Garson--Greer was her Irish mother's maiden name).
Greer was chosen much more recently by Kelsey Grammer for his daughter and by Brooke Shields in the Grier form. As a surname, it's associated with feminist writer/activist Germaine Greer.
Greer makes a good, strong, sophisticated and distinctive middle-name prospect.