Does this slang originate from half asked, since the difinition means exactly that. You only did half what I asked you.
By far the earliest instance of "half-assed" that a Google Books search turns up is from Thomas O'Brien & Oliver Diefendorf, General Orders of the [U.S.] War Department, Embracing the Years 1861, 1862 & 1863, volume 2 (1864), reporting on the court-martial of Captain John H. Behan on February 19, 1863:
Charge I.—"Conduct to the prejudice of good order and military discipline."
Specification 1st.—"In this ; that he, the said Captain John H. Behan, Company F, 16th Regiment Virginia Volunteers, while on duty in camp, on or about the 12th day of December, 1862, did use abusive and grossly insulting language to Joseph B. Hamilton, 2d Lieutenant of said Company F, before and in the presence of said Company F, while he, the said Joseph B. Hamilton, was on duty and was acting Adjutant of said 16th Regiment Virginia Volunteers, in words as follows, to wit: 'There goes our half-assed Adjutant ;' which was calculated to impair and weaken the influence and control of said Lieutenant Joseph B. Hamilton as Adjutant of said regiment, and also his influence and control over said Company. All this at or near Miner's Hill, Virginia, on or about the 12th day of December, 1862."
The captain was found Not Guilty on this charge, but Guilty on unrelated charges of having knowingly accepted a stolen sword from another soldier and of having refused to return to yet another soldier a sum of about $34 placed in his trust by that soldier.
The next instance of half-assed/halfassed that the search finds is from 1934, in Josephine Herbst, The Executioner Waits, where the closed-up form occurs at least twice. Here is one of them [snippet]:
He hardly listened to Jonathan until he caught the words, "And what I'm going to do is just light out, go to New York. I'm sick of these halfassed towns."
Numerous instances of half-assed turn up in search results from the late 1930s forward.
The first instance of half-arsed/halfarsed that the search turns up is from John Simon, Movies into Film: Film Criticism, 1967–1970 (1971) [series of snippets]:
The story, as everyone knows, concerns Myron Breckinridge, who, after a Scandinavian operation, becomes Myra. Equipped with physical beauty, knowledge from within of both sexes, and expertise in Hollywood lore, she sets out to capture both the men and the women of filmland today, and tomorrow the world. The movie turns all this into a dream, and cleans up the comically lewd incidents by reducing comedy to oafishness and lewdness to suggestive smirking. That kind of deliberately halfhearted — or, in this context, halfarsed — cleaning up is the real dishonesty, the real smuttiness of the enterprise. For the sexual acts are now performed by half-clothed Barbie dolls with carefully castrating camera angles, and if the dirtiness is not for real, it must be for dirty.
Interestingly, three useful collections from the period 1890–1915—Farmer & Henley, Slang and Its Analogues, Fla–Hyps (1893), Barre & Leland, Dictionary of Slang, Jargon & Cant, A–K (1897), and Thornton, An American Glossary, A–L (1912)—have no entry for either half-assed or half-arsed, despite the fact that the term clearly had been in use since at least 1862. Farmer & Henley is by no means squeamish about reporting on naughty words, so I find the term's absence there particularly baffling; I don't know how much the other two books may have been influenced by considerations of propriety.
Merriam Webster's Online, by the way, traces half-assed only to "circa 1932." Can the term really have gone underground for 70 years?
This might not explain its origin as much as its popularity, but as I noted in a comment, "half-assed"/"half-arsed" is of a piece (groan ;)) with lots of other idioms in which
inadequacy = not having one's full complement of body parts, as in
She's cruel; she has only half a heart; she's heartless.
He's got half the brains she does.
He's a eunuch./Grow a pair!
There is also
inadequacy = not having one's body parts properly arranged, as in
That whole thing's ass backwards (or bass ackwards.)
He has two left feet.
Her head's not screwed on right. etc.
The origin may have been covered in other answers, but maybe not the meaning.
The “half-assed” part refers to having your pants not done up all the way yet, with half your ass showing. If you are in that state, you are likely to be unprepared for the day’s tasks and generally incompetent.
So the admonition:
Don’t go off half-assed.
… reminds you to stop, prepare, then go off and do a task successfully.
Within a translation of a greek joke book (Philolegos: The Laugh Addict) estimated to have been written around 400 CE, there is a joke that paraphrased goes like this: An Abderite man who owes someone an ass but doesn't have one to pay his debt asks if he can give two half-asses instead.
Based on the other jokes about Abderites they are from a particular region of Greece whose residents were portrayed as stupid and deformed, similar to cretins. If I understand the joke it is that this person does not own an ass but is so associated with half-assedness that he has plenty of that to pay with. If that is a correct interpretation then the term half-assed was in common use since at least the late period of the Roman empire.