I'd also be interested to know how some NRL teams got their names
Most of the older teams existed without a mascot name for decades. In some cases a nickname developed naturally, while in other cases a name was chosen to match their long established colours. Balmain's 1908 logo featured a kangaroo and a tiger supporting a football in the style of a coat of arms. Wests, in their black and white, adopted the Magpie while they were still based around Ashfield as it matches their colours and is ubiquitous in the suburbs.
Most of the more modern teams held competitions to come up with their names, though most of them ended up adopting the names of American sports teams. The Raiders, Steelers, Broncos and Cowboys all copied their names from NFL teams, though in some cases there was a genuine reason why. The Steelers, for example, is as perfect a name for a team from the steel-producing city of Wollongong as it is for Pittsburgh. Canberra claimed they chose their name because they were going to "raid" Sydney, but in reality they just copied the Oakland Raiders name and then "raided" the Brisbane competition to fill out their playing roster. Some older teams that had not developed a mascot name also used this approach. Newtown copied the New York Jets, discarding the nickname "Bluebags" that fans had been using for decades.
Souths probably have the most interesting story. The South Sydney players during the great depression used to make ends meet by skinning and selling rabbits from carts in the streets of Darlinghurst, Surry Hills and Redfern. They would play in the blood-stained jerseys on the week-end and opposing players didn't want to tackle them. The South Sydney crowd would taunt the away team by chanting "Rabbitoh".
Sydney/Easts copied their "Roosters" name from a visiting French national side (who also wore the tricolour) and believe it is ideal because the Eastern Suburbs see the sunrise first.
Norths got "Bears" from a local shopping centre that sponsored them.
St George were sometimes called the "Illawarra-ites" and "Illawarra-men" in the 20's and 30's
I don't remember the origin of Penrith's name, but I imagine it comes from the urban myth that there is a panther roaming the Blue Mountains area. Sightings of the panther date back over a century.
The Bulldogs didn't call themselves that until the 70's and had been informally referred to as the "Berries" before that.
Parramatta means "where eels live" or something like that in the local Aboriginal dialect. So it seemed like the obvious choice when everyone was getting mascots in the 70's. Before that they had been known (like Cumberland before them) as the "fruitpickers".
I once read a new article that described how all the teams got their names, but that all I can remember now.
Edit: Dammit! Ninja'd again. I must learn to type faster.