The Australian Aborigines did, and still do in many cases, call themselves by specific tribal names. When Europeans first settled Australia, there were somewhere around 500 different languages among the native peoples, and even now there are close to 200 still surviving.
This page gives the names and locations of some tribes, and if you want to do a comprehensive search you can check out the AusAnthrop Australian Aboriginal tribal database, where you can search by tribal name, state, or linguistic zones.
In general discussions about Australia’s first inhabitants, the word “Aborigine” is often used, with “Aboriginal” serving as the adjectival form. There are also regional terms which cover specific sections of Australia, but are not specific enough to deal with individual tribes. For example, the indigenous peoples of the states of New South Wales and Victoria often refer to themselves collectively as Koori. Other regional appelations include Murri (from Queensland) and Nunga (from South Australia). There are others, but i can’t remember them now. I’m sure that if you Google you could find out what they are.
I’m no expert on the naming practices of different Aboroginal tribes or groups, particularly their pre-contact practices. All i know is based on my experience hearing some Aboriginal names in the news, and knowing some Aborinial people when i lived in Australia.
The names of Australian Aborigines in modern Australia constitute a rather diverse group. Many have names that are indistinguishable from Europeans, some have obviously Aboriginal names, and some combine European and Aboriginal names.
For example, there’s no way that you could tell by her name alone that political activist and former magistrate Pat O’Shane is an Aborigine. Same with athlete Cathy Freeman.
Galarrwuy Yunupingu, on the other hand, would be instantly identified by most Australians as an Aborigine, based on his name alone.
Neville Bonner’s name seems rather European. Actor David Gulpilil has a European first name and an Aboriginal last name, as does Aussie Rules football player Gavin Wanganeen.
One of Australia’s most famous tennis players had a European first name and an Aboriginal last name. When she married, she kept her last name, but added her husband’s name to the end to become Evonne Goolagong Cawley.