Ah, speculation and folk etymology! Since on guess is as good as another ... I postulate:"Heave away your ol' linking"
Your ol'/old' is simple enough; as for linking -- well. those lines/hawsers/ropes they were hauling were obviously linked to something.
And our friends in Oz can discard out-of-date blue clicky things as they're bound for South Australia.
But let's get to the real question to be deconstructed here:
In the first 'verse' the chanteyman claims to have been born in South Australia, but down near the end he asserts that he is 'Liverpool born and bred'. Aha!