Here is the part on the emblem (Wikepedia):
Crest
The original Rabbitohs emblem introduced in 1959
One version of how the club got the "Rabbitohs" nickname comes from their
rugby union days at the turn of the 20th century. In those hard times, players wearing their
cardinal red and
myrtle green football jumpers, earned some extra money on Saturday mornings by hawking rabbits around the district with the traditional cry of "Rabbitoh!" echoing through the narrow streets.
[9]
As they made a sale, they would sling the bunny from their shoulder and skin it on the spot, inevitably accumulating some of the fur and blood on their jerseys as they did so. When they played in those blood stained jumpers that afternoon, opponents from wealthier rugby clubs did not always appreciate the aroma and would mockingly repeat the "Rabbitoh!" cry.
[10]
Another account of the legend relates that the Rabbitoh name was a disparaging reference by opposing teams to South's home ground being plagued with "rabbit 'oles". In those early days
Redfern Oval was then known as Nathan's Cow Paddock.
[2] Yet another version links the Rabbitoh name as being adopted from that of the touring
Australian rugby union teams of the early 1900's who where nicknamed "Rabbits" prior to discarding the name in 1908 in favour of the moniker "Wallabies".
[11]
The "Rabbitoh" emblem (a white running rabbit) first appeared on the team's jersey in 1959. The Rabbitoh emblem has in various forms been carried as the club's crest on all players jerseys ever since. The original "Rabbitoh" emblem design that appeared on the team's jerseys throughout the 1960's and 1970's has now been incorporated on the current jersey.