There is nothing to say whether she was an 'only child' or not. However, the Walhalla Heritage and Development League web site shows a number of other 'Barretts' in Walhalla during the same period. The odds of at least some being related are high.
Alfred John Barrett (Painter) m. Harriet Mary Brown (2 children).
Edna Muriel Barrett b. Walhalla 1905
Elizabeth Kate Barrett m. James Robert Hawkins (4 children).
Jack Barrett b. Walhalla 1908
Jane Matilda Barrett m. James Lindrea (2 children).
Joseph O'Hara Barrett b. Walhalla 1872 m. Ada Gardener
Mary Ann Barrett m. Richard Rawson (4 children).
Michael Barrett (Miner) m. Catherine Longey (1 child).
Phillip Barrett (Painter) d. Melbourne 1915
Rosenia Barrett m. Alfred Joseph Watson (1 child).