Welcome to Rootschat, drifter
If your family has an unusual surname you may be able to find some clues through the telephone listings:
http://www.whitepages.com.au/search/residential?subscriberName=&location=You can also look them up on the National Archives site which will probably have their emigration details - link in the article JM mentioned. This should at least tell you which State they were ticketed to; obviously they may not have stayed in that State. Search the National Archives site as a guest.
You can then look at the various State BMD lists for deaths - eg the NSW BMD site has deaths up to 1980 (BMD registrations are handled by each State rather than nationally). The Ryerson index has lists of deaths up to very recent but it is not a comprehensive list.
http://ryersonindex.net/search.phpA search of TROVE may turn up family notices such as engagements, marriages, deaths etc but most digitised newspapers are only on-line up to 1954.
http://trove.nla.gov.au/newspaper10pound Poms may have been 'sponsored' (had a job to go to, or friends/family to start off with) or may have found their own way in accommodation, job-hunting etc. There were hostels also which new arrivals could go to.
I am happy to do look-ups on the electoral roll for you but you will have to make 2 more posts on Rootschat before you can PM me with the names. It will need to be done by PM in case there are still living members of the family.
Good luck with your search. Judith