Why are crazy people called nut jobs? Why are they called a job? Wiktionary is of no help here.
Maybe looking at its origin will help. SlangDictionary.com defines "nut-job" as:
a more recent variation of nut-case
a crazy person. A slang version of ‘mental case’ which spread from American speech into British usage at the end of the 1950s.
Thus, if "nut-job" was a mere variant of "nut-case", then, "job" would make sense. A "case" has two meanings:
a person or thing whose plight or situation calls for attention:
A piece of work, specifically defined within a profession.
The second meaning of "case" is synonymous with "job", and thus, in when "nut-case" was given a variation, its synonym "job" was taken and used. Thus, there would be a logical reason why "job" is used here.
This use is very similar to whack job and also similar to other slang uses of job such as "do a job on" or "a bank job" (meaning to rob a bank). Here, job just means something of a particular type.
"I got a new oven, one of those self-cleaning jobs."
"Use a real-dictionary, not one of those new-fangled online jobs."
You'll also find nut job used a lot in this same construction. "He's one of those right-wing conspiracy nut jobs."
I knew the etymology of "nut" from OED in that it can refer to the head. Etymology of "job" refers to "piece of work" from the 1550s (http://www.etymonline.com/index.php?term=job). I would guess the way that we slangily use "piece of work" to mean someone who doesn't make sense, "nut job" would logically follow as it is shorter than "nut piece of work"...
As others have pointed out, this usage of "job" basically derives from "piece of work". It's not normally used of people, but it's been around a long time in slang. More commonly in the diminutive form jobbie, as in "A Blackberry is one of those hand-held jobbies".
All job really means here is "example" ("instance of"). In this slang usage there's no particular implication that the thing (or person, here) being referred to is either a task, or the end result of task-oriented labour.
Anyway, here's a chart to clarify that nutjob isn't really going anywhere...
I'm a firm believer many words like these function as compound words. I would think to define the term by it's constituent parts.
Thus, a definition of a nut job could be a label for a person who acts nutty, loony, weird, "out of touch with the main-stream". Plus this behavior is burdensome to the people who interact with the person--exhausting, demanding, unavoidable, persistent, etc. Its work to be their friend or acquaintance.
What makes the phrase interesting it the word "job" may seem to refer to the person defined by their actions. The uses I've heard (and found below) are not synonymous with "nutcase". Again, using the compound word comparison, "nut case" would be:
loony person + document case file = certifiable crazy person.
While "nut job":
loony person + diminutive work = someone made crazy by their work. Here are two quick examples of politician and actor. Oddly, the male examples I found tended to use nutjob with examples of extreme antisocial behavior (criminals).
Maybe nutjob is a gendered slang term?