Labor

I would have bet that the Australian English spelling would be like the British English spelling, since Australian English tends towards the British English end of the spectrum rather than the American English. Especially since names tend to persist, and it's probably been around for a while.

goes to check Wikipedia to see whether it was renamed

Interesting. Not exactly. The article uses "labour", and has a section dealing specifically with this:

https://en.wikipedia.org/wiki/Australian_Labor_Party

In standard Australian English, the word labour is spelt with a u. However, the political party uses the spelling Labor, without a u. There was originally no standardised spelling of the party's name, with Labor and Labour both in common usage. According to Ross McMullin, who wrote an official history of the Labor Party, the title page of the proceedings of the Federal Conference used the spelling "Labor" in 1902, "Labour" in 1905 and 1908, and then "Labor" from 1912 onwards.[11] In 1908, James Catts put forward a motion at the Federal Conference that "the name of the party be the Australian Labour Party", which was carried by 22 votes to 2. A separate motion recommending state branches adopt the name was defeated. There was no uniformity of party names until 1918 when the Federal party resolved that state branches should adopt the name "Australian Labor Party", now spelt without a u. Each state branch had previously used a different name, due to their different origins.[12][a]

Although the ALP officially adopted the spelling without a u, it took decades for the official spelling to achieve widespread acceptance.[15][b] According to McMullin, "the way the spelling of 'Labor Party' was consolidated had more to do with the chap who ended up being in charge of printing the federal conference report than any other reason".[19] Some sources have attributed the official choice of Labor to influence from King O'Malley, who was born in the United States and was reputedly an advocate of English-language spelling reform; the spelling without a u is the standard form in American English.[20][21]

Andrew Scott, who wrote "Running on Empty: 'Modernising' the British and Australian Labour Parties", suggests that the adoption of the spelling without a u "signified one of the ALP's earliest attempts at modernisation", and served the purpose of differentiating the party from the Australian labour movement as a whole and distinguishing it from other British Empire labour parties. The decision to include the word "Australian" in the party's name, rather than just "Labour Party" as in the United Kingdom, Scott attributes to "the greater importance of nationalism for the founders of the colonial parties".[22]