Some points of interest from the OED entry. The earliest cite is this one:
1580 C. Hollyband Treasurie French Tong Le hoquet, the hickop, yexing.
Yoking, yelking or yexiing is West Country dialect for hiccuping although I’m not sure it’s still current. I was born in Devon and have many relatives there but I can’t say I’ve ever heard it. OED is unusually stern on the ludicrous hiccough form, which surprisingly goes back to at least 1626 (Francis Bacon in his posthumously published work Sylva Sylvarum.
Hiccough was a later spelling, apparently under the erroneous impression that the second syllable was cough, which has not affected the received pronunciation, and ought to be abandoned as a mere error.
That’s an 1898 entry but I see from Google that hiccough is still grimly hanging on, 411,000 hits to hiccup‘s almost 79,000,000.
This 1993 draft addition interested me too, not so much the definition - ’fig. A sudden brief or minor interruption in the normal progress of something; a hitch, setback; a decline in (esp. financial) performance which is assumed to be temporary.’ - but the first cite thereunder.
1965 Britannica Bk. of Year (U.S.) 869/1 Hiccup, adj. Informal. Using a direct dramatic opening for a movie, so that continuity must then be interrupted for the title and credits.
I’ve never seen that specific usage.
Ediit: Great thing of us forgot, I neglected to give the etymology. The word is from the earlier hickock, hicket, of which OED says:
One of the earlier forms of hiccup, the other being hickock, both apparently with a diminutive formative -et, -ock. The echoic stem hick appears also in Middle Dutch hick, Dutch hik, Low German hick, Danish hik, Swedish hicka hiccup, Middle Dutch hicken, Dutch hikken, Danish hicke, Swedish hicka to hiccup; also Breton hok, hik (Littré), French hoquet (15th cent.), Walloon hikéte, medieval Latin hoquetus (Du Cange), hiccup, French hoqueter (12th cent. in Hatzfeld & Darmesteter) to hiccup. The English hicket corresponds in formation to the French, and is identical with the Walloon. Assuming this to be the earliest form, we have the series hicket, hickot, hickock, hickop, hiccup (hiccough).