The only thing that I know is that the three of them lived in the Brisbane Cooparoo area in 1896-1899, though perhaps she wasn't living with her parents.  Around 1900 Lillian moved to Sydney, though I don't know how long she remained there.  I have no information on possible siblings or if Lillian even had a middle name.  There is a sense that she lived a rough childhood in the outback, which makes me think that her father might have been a grazer at one point, but it is only a guess. 
I hope some of that helps.  That is truly all the details that I have.....  other than I also found that her father had a drinking problem.  
Thanks beforehand for anything that you might find.
Kathy