Did you gggrandmother have any siblings?
I see that George Brockenborough and Sarah ( no surname given for her) had 3 children registered -
George, 1877 registered at Walgett (Edit: died 1877, reg Walgett, surname Brokenborough)
Margaret Florence, 1879, reg Walgett
Hannah Elizabeth, 1881, reg Bourke (Edit: died 1881, reg Bourke, surname Brokenborough)
There is a public tree which suggests that George Brockenborough's wife and mother of these children was Sarah Margaret McCusker and adds another child James William 1883, reg at Walgett - spelt BROKENBOROUGH: . There are some older half-siblings for the Brockenborough children.
It is not looking as though this is your gggrandmother - but it sure is a co-incidence. Please let us know what you find on the birth certificate.
Judith