The most recent xkcd:
This is such a good joke, and in retrospect such an obvious one, that it's hard to believe that Randall Munroe was the first one to tell it — but I can't find any precedents. Of course, "significant other" has only been in common use since the 1970s.
In a scientific context, plain "significant" is usually interpreted to mean "statistically significant" — this might be one of the most important lexicographical developments of the 20th century. I believe that the credit, or blame, goes to the public-relations genius of Ronald Fisher (1890-1962), exemplified in his brief popular article, "Mathematics of a Lady Tasting Tea". (For some relevant discussion in earlier LL posts, see "Listening to Prozac, hearing effect sizes", where "statistically significant" is contrasted with "clinically significant"; or "The secret sins of academics", which includes some fun quotes from Deirdre McCloskey on the perils of significance testing without a loss function; or the series of posts beginning with "The 'Happiness Gap' and the rhetoric of statistics" and ending with "The 'Gender Happiness Gap': statistical, practical and rhetorical significance".)
But the female lead in this xkcd strip makes her point, not with a t-test or some other method from inferential statistics, but rather with a boxplot, and the common-sense assertion that "you spend twice as much time with me as with anyone else". Boxplots (R help page here) are a method of "exploratory data analysis" (EDA), invented by another great public-relations genius of 20th-century statistics, John Tukey (1915-2000).
Alas, even a combination of inferential and exploratory statistics with clever wordplay is not enough to guarantee happiness: the strip's mouseover title is "… okay, but because you said that, we're breaking up." Here's hoping that the male character gets over his (feigned?) commitment issues.
Among John Tukey's best-known coinages, outside of the realm of EDA, are the terms "bit" and "software". My personal favorite, though, is his term for the Fourier transform of the log of the squared magnitude of the Fourier transform of a signal — since this is the spectrum of the spectrum, in some sense, he suggested that it should be called the "cepstrum". And since a Fourier transform takes us from time to frequency, a second one obviously takes us from the frequency domain to the quefrency domain. And since a filter enhances or attentuates frequency components, an operation that enhances or attentuates quefrency components must be a lifter.
This is the only example that I can think of where scientific or mathematical terminology was created by spoonerism.
For various interesting reasons (for example, when two signals are convolved, their cepstra are added), the cepstrum is in fairly wide use — the standard acoustic parameters for speech recognition are mel-frequency cepstral coefficients (MFCCs), for example.
[Update: in addition to the 3/7/2008 webcomic precedent found by atta in the comments, Victor Steinbok sent email citing a Yelp page from December 2007, an ask.metafilter.com page from January 2006, and a blog post from 2/1/2002, which all deploy the "statistically significant other" phrase in one way or another. (The 2002 blog post: "My statistically significant other comes in with p < .001, so I think I'll keep him.") This is the result of Victor's exhaustive search of the Google hits for the phrase. Considering the reflex association significant → statistically significant, this is a remarkably small number of precedents -- I guess it reflects exactly the compartmentalization of mathematics and romance that xkcd so strikingly overcomes.
Victor also notes that
... "insignificant other" has earned an entry in the Urban Dictionary. It's also a part of a title of a movie, two books, at least one song and a YouTube video. There is a cafepress "license plate frame" and an IO abbreviation is listed in TheFreeDictionary.com.
There's no report yet on "clinically significant other"... ]