mmm, praps.  ;)
I had understood it to mean upset, perhaps similar to ‘roil’; David Crystal glosses it as ‘Turmoil, disturbance, fuss ~ This mortal coil – ie, the bustle of life…’
I have no idea if it was a Shakespearean coinage or antedated, but he certainly uses it elsewhere.
For example, in 
The Tempest (Act 1, Scene ii):
Ariel:
To every article. 
I boarded the king's ship; now on the beak, 
Now in the waist, the deck, in every cabin, 
I flamed amazement: sometime I'ld divide, 
And burn in many places; on the topmast, 
The yards and bowsprit, would I flame distinctly, 
Then meet and join. Jove's lightnings, the precursors
O' the dreadful thunder-claps, more momentary 
And sight-outrunning were not; the fire and cracks 
Of sulphurous roaring the most mighty Neptune 
Seem to besiege and make his bold waves tremble, 
Yea, his dread trident shake.
Prospero:
My brave spirit! 
Who was so firm, so constant, that this coil 
Would not infect his reason? 
He also uses it in 
Much Ado About Nothing [Act 3 scene iii]: Dogberry is charging the officers of the watch he has sworn in…
One word more, honest neighbours. I pray you watch
about Signior Leonato's door; for the wedding being
there to-morrow, there is a great coil to-night.
Adieu: be vigitant, I beseech you.
I think ‘coil’ also crops up in 
Two Gentlemen of Verona somewhere in the first act and possibly one or two of the other plays, with a pretty uniform sense of ‘disturbance’, of a wild edged kind.
Edit: yes, here’s a list of usages in this pattern:
http://www.opensourceshakespeare.org/search/search-results.phpI notice MW gives an unhelpfully brief "origin unknown" :(