01/01/1901 Federation of Australian Colonies (i.e. date of proclamation by the first Governor General. The Commonwealth of Australia Constitution Act 1900 (UK) was passed on 05/07/1900 and given royal assent by Queen Victoria on 09/07/1900.)
First January is a good date for a national day, because Australians would celebrate their nationalism while nursing a hangover, as would be entirely apt†. OTOH 01/01 is already a holiday.
†We're a nation of fecking dipsos, is the point I'm making here.
11 December
11/12/1931 Statute of Westminster Act passed by UK parliament (date of royal assent) giving British dominions legal independence from UK parliament.
Not ratified by Australia's lickspittle imperialist government for eleven years. Bit close to Christmas. Not really Australian.
9 October (3 September?)
09/10/1942 Statute of Westminster Adoption Act 1942 passed (royal assent - commencement retroactive to 03/09/1939 [outbreak of World War II])
4 December (17 February?, 3 March?)
04/12/1985 Australia Act (date of royal assent [GG] - ratified by UK Parliament [royal assent by ERII] 17/02/1986 - commenced 03/03/1986)
Yay! No longer a vassal state of the British empire. Now we're free to focus on being a vassal state of the American empire!
Local assent is a bit close to Christmas (OTOH from Easter to Christmas is a bit of a public holiday wasteland in Oz, depending which state you're in); commencement date is close to Labour Day (and variants) in Vic, Tas, WA (very), and Adelaide Cup Day - also we only had to delay the commencement date so that the Brits had time to say "yeah, alright then" which also seems odd to celebrate.
As I tell people when they ask, I would prefer to leave the flag, anthem and date of Australia Day unchanged, because it is so helpful in impressing on people, especially young people, that nationalism is childish and stupid, to have a duff national flag, a duff national anthem, and an entirely duff national holiday.
If we do change the flag, it should be replaced with the Eureka design, and people can choose their own colours.
*...s 4 of the Statute only affected UK laws that were to apply as part of Australian Commonwealth law, not UK laws that were to apply as part of the law of any Australian state. Thus, the Parliament of the United Kingdom still had the power to legislate for the states. In practice, however, this power was almost never exercised. For example, in a referendum on secession in Western Australia in April 1933, 68% of voters favoured seceding from Australia and becoming a separate Dominion. The state government sent a delegation to Westminster to request that this result be enacted into law, but the British government refused to intervene on the grounds that this was a matter for the Australian government [after 18 months of committee hearings]. As a result of this decision in London, no action was taken in Canberra or Perth. - Wikipedia.(Amendments 26/01/2023 as indicated, including second footnote.)
**According (again) to Wikipedia: "The immediate prompt for the adoption of the Statute of Westminster was the death sentence imposed on two homosexual Australian sailors for the murder of their crewmate committed on HMAS Australia in 1942. Since 7 November 1939, the Royal Australian Navy had operated subject to British imperial law, under which the two men were sentenced to death. It was argued that this would not have been their sentence if Australian law had applied, but the only way for the Australian government to get the sentences altered was by directly petitioning the King, who commuted them to life imprisonment. Adopting the Statute of Westminster, so that Australia became able to amend applicable imperial law, avoided a potential repetition of this situation. The men's sentences were later further reduced."