Mark Liberman's post on a recent xkcd strip unleashed a flurry of comments about jokes that follow the template, "X-er? I hardly know 'er!" (The strip used "supercollider" in the template, an apparent homage to "Futurama.") Commenters were also reminded of a somewhat similar bit of musty British humo(u)r:
A: My wife's gone to the West Indies!
A: No, she went of her own accord!
The success of the joke, such as it is, requires being able to interpret [dʒə ˈmeɪkə] as a clipped form of "Did you make her?" As I discuss in the post "Pinker's almer mater," Led Zeppelin alluded to this joke by titling a reggae-influenced song, "D'yer Mak'er" (recorded in 1972, released the following year). This non-rhotic pronunciation spelling is utterly lost on most (rhotic) American fans, who would likely be puzzled by the original joke anyway.
David Eddyshaw recalled hearing the "Jamaica" joke in the mid-1960s and suggested it might go all the way back to the music-hall era. I did some poking around and discovered a couple of cinematic appearances. The first is in the 1955 film "The Colditz Story," about Allied POWs in World War II imprisoned in a German castle. The prisoners put on a musical revue, and two of them, played by Ian Carmichael and Richard Wattis, do a comedic routine with well-worn jokes about Venetian blinds and Greek urns. Through the magic of YouTube, here is the routine, with the "Jamaica" bit at the end:
The joke also shows up in "The Young Ones," a 1961 musical starring Cliff Richard. It's your basic "let's put on a show" movie in the tradition of Mickey Rooney and Judy Garland, and the "Jamaica" joke is told as part of the aptly titled production number, "What Do You Know We've Got a Show." Once again, YouTube comes to the rescue — the joke appears at the 3:00 mark:
Since both of these films pay homage to the "I say, I say" routines of music hall "double acts," the joke is no doubt rather ancient, as David Eddyshaw conjectures. How exactly this golden oldie relates historically to the "hardly know 'er" genre remains to be investigated.
[Update #1: In the comments, Ray Girvan uses Google Book Search to trace the joke back to 1914, appearing in The Railroad Telegrapher, house magazine of the Order of Railroad Telegraphers. It shows up in a number of technical magazines after that.]
[Update #3: See this post for more on using Google Book Search (and now the Hathi Trust) for finding early examples.]