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And our friends in Oz can discard out-of-date blue clicky things as they're bound for South Australia.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!