What is the origin of the term 'Swinging the Lead'?

Naval term.

Swinging the Lead - (RCN) To work in a lazy manner.

The expression 'He's swinging the lead' comes from days before sonar was used to detect under keel depth. A man was placed forward and swung a lead weight with a length of rope. A difficult and tiring task, so seamen would often be seen from aft 'swinging the lead' instead of actually letting go.

Here's a little more to add from another site:

Swing the lead - shirk, skive or avoid work, particularly while giving the opposite impression - almost certainly from the naval practice of the 19th century and before, of taking depth soundings by lowering a lead weight on the end of a rope over the side of a ship. (According to etymologist Michael Quinion, the lead lump weighed nine pounds and had tallow - grease - on its base, which also enabled a sea bed sample to be brought up from below; the rope had colour coded markers to help gauge the depth.) It seems entirely logical that the impression would have stemmed from the practice of time-wasting while carrying out the depth soundings: a seaman wishing to prolong the task unnecessarily or give the impression of being at work when actually his task was finished, would 'swing the lead' (probably more like allow it to hang, and not doing anything purposeful with it) rather than do the job properly. A lead-swinger is therefore a skiver; someone who avoids work while pretending to be active.