If there was a newspaper notice of the death, it was probably in
the Illawarra Mercury (incidentally, one of Australia's oldest newspapers).
The site only displays classified notices from the last week but a Google search {site:illawarramercury.com.au archive 2010} generates plenty of hits indicating an extensive back catalogue.
Unfortunately, they all lead to 404 (Page not found) apparently as a result of building a "new" website.
The Ryerson Index_ is a community-generated index to death notices appearing in current Australian newspapers.
It also includes some funeral notices, probate notices and obituaries. The website allows you to check the extent of coverage of each newspaper.
Although the Mercury is incorrectly placed in some alphabetic lists as WOLLONGONG ILLAWARRA MERCURY, the coverage is excellent.
RANGE OF NOTICES INDEXED
Death Notices..... From January 1st, 1997 to November 10th, 2012 (40999)
Funeral Notices.. From January 1st, 1997 to November 3rd, 2012 (2197)
Use the surname and the presumed timespan to launch your search. With a little luck, you will locate a notice that refers to the deceased as "beloved sister of ..."
or something similar that will open up new lines of research.