This word is delightfully evocative, bringing to mind some languid person lolling on a couch while all around goes to ruin, so accurately evoking its modern idea of somebody who lacks enthusiasm and determination or is carelessly lazy.
It owes its origin, strangely enough, to an old saying of regret or dismay, lack-a-day!, a shortened form of alack-a-day! Alack dates back to medieval times, and probably comes from a dialect word lack that is variously interpreted as failure, fault, reproach, disgrace, or shame. So alack-a-day! originally meant “Shame or reproach to the day!” (that it should have brought this upon me). But over time it became weakened until it became no more than a vapid and vacuous cry when some minor matter went awry.
At some point in the eighteenth century, the form lackadaisy appeared, with lackadaisical coming along shortly afterwards for somebody who regularly used the cry. At first it meant that the person was feebly sentimental rather than lazy. The first person recorded as using it was Laurence Sterne, in his Sentimental Journey of 1768:
I took hold of her fingers in one hand, and applied the two forefingers of my other to the artery. — Would to heaven! my dear Eugenius, thou hadst passed by, and beheld me sitting in my black coat, and in my lack-a-day-sical manner, counting the throbs of it, one by one, with as much true devotion as if I had been watching the critical ebb or flow of her fever.
Later it moved towards the idea of somebody who was affectedly languishing, and thence to someone merely lazy.