The two specific dates that are directly tied to the establishment of the Commonwealth of Australia are "January 1st" and "September 17th". But, celebrations began on December 31 and the Inauguration of the new Commonwealth was celebrated in all towns and cities at midnight New Years Eve.

Any other dates randomly chosen would be just as meaningless as Convict Day (Jan 26).

Queen Victoria, Proclamation, 17 September 1900.

We do hereby declare that on and after the first Day of January one thousand nine hundred and one, the people of New South Wales, Victoria, South Australia, Queensland, Tasmania and Western Australia shall be united in a Federal Commonwealth under the name of the Commonwealth of Australia.

We should be careful with celebrating birthdays, look what happened with the Queens Birthday, a particularly difficult delivery. It seems it all started on October 2nd, most of us just call that Labour Day, that's when labour started,. except the silly cane toads, who thought it was an easy birth and her majesty popped out then, they all went off to the pub for a XXXX and haven't returned since, the cane toads are still wrongly celebrating October 2nd, as the Queens Birthday, they have been having such a good time of it, no has the heart to tell em' they got it wrong . Not so, seems the West Australian's had detected labour pains a week earlier on September 25th and thought here she comes, unfortunately a false alarm, but no one told the sand grouper's, so they have been blissfully celebrating September 25th as the Queens Birthday ever since. In the rest of Australia we waited patiently for an announcement from Buck House, we waited and we waited, the announcement came via Captain Cook, June 12th, that the Queen Mum, that's the mother of the Queen for those who don't know, had, had a bouncing baby girl. Something was lost in the translation, it was probably due to Cook's pommy accent. when we found out that she had actually been born April 21st, we were all ready to celebrate! Until some bright spark pointed out April 21st 1926 had been a Wednesday. We all looked at each other, and declared we didn't care what day she was born, but we are having a day off on a Monday!
I actually favor Monday 13th March for the Queen Birthday holiday, March, nice time of the year. Unfortunately March 13th is Adelaide Cup Day, and I don't want to upset anyone who is celebrating something of more importance and significance than the Queens Birthday, like the Adelaide Cup.