Hi there Paul,
Perhaps you have already tried the NSW SRO Archives Investigator’s option for Probate Packets at this link
http://investigator.records.nsw.gov.au/Search.aspx I chose the simple search option and entered “Eliza Jane death” for the keywords. There’s a fair few results come up, but I have chosen one as an example to follow up. I chose Eliza Jane Howard, date of death 9 April 1941 with probate granted 19 June 1941. (ie if that would be your elusive lass, then she would have been around 75 years of age)
1. Found the likely NSW BDM registration indexed on line. Her mother was recorded as Eliza (although her father was recorded as Thomas and not as William). It was registered at Mosman. #9409
2.Found the newspaper cutting online
http://trove.nla.gov.au/ndp/del/home in the SMH 10 April 1941. That lady was a widow, and a married daughter is named in the cutting. I won’t mention the daughter’s name here because she may be living. The Funeral notice (same issue of the paper) shows the funeral at St Lukes at Mosman and interment at Randwick. There would be other closer cemeteries to Mosman so it is likely that Mrs Howard’s wishes were for interment at Randwick.
3. If this were your elusive lass then I am sure you would be looking for headstone inscriptions and grabbing a plastic card to order a transcript of that d.c. ....
I chose that particular one to follow up because to me, the granting of probate within eight or nine weeks after death often indicates a small estate or an uncomplicated will, and afterall I am trying to help find your elusive Eliza Jane, perhaps you may be able to use that example to find either her, or perhaps other family members.
Edit to add Easter was 13 April in 1941, so there would have been several public holidays to observe and courts would have closed for those days, thus delaying any probates already in the system and possibly a backload so my thoughts re small or simple estate could be on track.
Cheers, JM