Hi Ian,
Do you know where in Wicklow your relatives may be from? Do you have any references to a village/town or other placename?
Have you checked the immigration records in Australia? Generally, more information was given at the port of arrival rather than the port of departure. The University of Woolongong has produced, on microfiche, a complete index and transcript of all information concerning immigrants of Irish origin recorded on ships' passenger lists between 1848 and 1867. These are useful for finding out an exact place of origin as well as parents' names. The Public Record Office of Victoria has good online databases of settlers at www.prov.vic.gov.au Otherwise, other records may be found in the Colonial Office Papers of the UK National Archives, class reference CO 201. This class contains a wide variety of records, including petitions for assisted passages, emigrants' lists, records of emigrants on board ship, petitions from settlers for financial assistance and much more.
You may also try Church of Ireland parish registers for the period up to 1870 which are public records. Registers are available for about one third of the parishes, however many were destroyed in the Public Records Office in Dublin in 1922. Most are still held by the local clergy, although some are in the National Archives of Ireland and others are in the Representative Church Body Library in Dublin. A list of all surviving registers is available in the National Archives. http://ireland.anglican.org/about/42 and http://www.nationalarchives.ie/.
You might try locating some members of the family via checking the land records called the Tithe Applotment Books (1823-38) or the later Griffith's Valuation (1848-64). The Tithe Applotment Books (1823-38): Microfilm copies of the books for all of Ireland are available at the National Archives of Ireland (NAI) http://www.nationalarchives.ie/genealogy1/genealogy-records/tithe-applotment-books-and-the-primary-griffith-valuation/ or the Church of Latter Day Saints (LDS) . Griffith's is freely available here: www.askaboutireland.com or here: www.failteromhat.com Failte Romhat has lots of other useful links you could try looking at.
Some records for Co Wicklow are available for a fee from http://wicklow.rootsireland.ie/
Kind regards,
Genealogy Support
Ireland Reaching Out