Could the BE slang term ‘kip’ meaning to sleep be a borrowing from Hokkien? I searched Hobson/Jobson and came up with this:
CHOP-CHOP . Pigeon-English (or -Chinese) for 'Make haste! look sharp!' This is supposed to be from the Cantonese, pron. kăp-kăp, of what is in the Mandarin dialect kip-kip. In the Northern dialects kwai-kwai, 'quick-quick' is more usual (Bishop Moule). [Mr. Skeat compares the Malay chepat-chepat, 'quick-quick.']
The characters here, as Janus suggested, are most likely 急急. Hobson-Jobson is clearly mistaken in referring to ‘kip-kip’ as Mandarin rather than Hokkien. This entry might be read as saying that ‘chop-chop’ and ‘kip-kip’ were competing forms donated by different Chinese dialects/languages. However, it’s not clear if the latter was ever used in English or not.
In any case, if English had already borrowed a version of this Chinese word to mean ‘quick’ with no semantic shift, why would it re-borrow the same word from a different dialect to mean ‘sleep’? It seems highly unlikely. Generally, to establish a relationship of borrowing we would want to have three things: (A) a phonetic link, (B) a semantic link and (C) a context (a time and place where the donor and recipient languages would have been in contact.)
With ‘kip’ we have A, it seems, but even that could be challenged. Initial ‘k’ represents an aspirated stop in both English and pinyin, but in other Chinese dialects it might well be pronounced without aspiration and thus sound more like an English ‘g’. And how did Cantonese ‘gap’ become English ‘chop’? Sound change in borrowed words can be idiosyncratic.
Concerning B, any semantic link between ‘quick’ and ‘sleep’ would be forced. Did Chinese people in a 19th century treaty port who wanted to take a nap after lunch ever say ‘kip-kip’ to signify ‘just a short one’ to English people, who then misinterpreted the word to mean sleep? It seems too tenuous to take seriously.
Finally C, the context. Where would the hypothetical encounter in B have taken place, and at what point in time? The history of Chinese-Western contact is of course very complex. We might be talking about the south China coast in the 19th century. The main treaty ports frequented by English traders, at least before the country was ‘opened up’ after the 1840 Opium War, were Canton and Hong Kong. Since these are located in a Cantonese-speaking area, it’s probably safe to take Cantonese as the default donor language for loans into British English. However, other points of contact like Singapore would have had a different mix of dialect groups – generally for Southeast Asia the Hokkien communities were larger and more numerous than Cantonese ones, so the contact language might have been Hokkien. And so on.
The formal title of Hobson/Jobson is ‘Glossary of Anglo-Indian Words and Phrases…’, but it actually covers ‘Oriental terms’ more broadly, not just South Asian ones. More about this dictionary here:
Hobson-Jobson definitively
TL/DR: No, ‘kip’ wasn’t borrowed from Chinese.
Best Answer
When did the expression arise in its idiomatic sense?
Christine Ammer, The American Heritage Dictionary of Idioms ((1997) has this entry for make do:
Ammer is certainly correct about the sense of this set phrase, but make do (often followed by the preposition with) goes back at least to the middle of the nineteenth century in the United States and in Australia. Here are the first eight instances returned in an Elephind search of early newspapers. From the Keowee [Pickens Court House, South Carolina] Courier (August 4, 1849):
From "Greenhouse in Winter," in the [St. Clairsville, Ohio] Belmont Chronicle (March 18, 1853):
This example is interesting because the sense here is "make them do"—with the pronoun dropped.
From "Commissariat," in the [Perth, Western Australia] Inquirer (September 21, 1853):
This example—the first from Australia—is also a bit out of the ordinary for the set phrase; it seems to refer to "skins" and to mean something like "serve in the place of leather aprons." Ace researcher JEL notes in a comment below that "make do." here may not mean "make do" at all but may instead be an abbreviated form of "make ditto"—so that the line has the following sense:
Strengthening JEL's contention is the fact that none of the other items in the lengthy commissariat list ends with a period before em dash and following quantity.
The remaining five examples, however, possess the same sense of the phrase as the 1849 example I began with, which is (to repeat Ammer) "get along with the means available." From the "The Wheat Buyers Attorney," in [Red Wing, Minnesota] Goodhue Volunteer (July 2, 1862):
From "Water Police Court: Violent Death of Henry Kinder: Charge of Murder," in the Sydney [New South Wales] Morning Herald (December 7, 1865), page 6:
From a letter by W. C. Sanders to the editor of the [Adelaide] South Australian Register (February 22, 1871):
From a letter by Thos. Hardy to the editor of the [Adelaide] South Australian Register (December 1, 1874):
From "General News," the [Balranald, New South Wales] Riverina Recorder (January 18, 1888):
Where did it come from?
One possible source of "make do" is as a short form of "make it do." Writers have used "make it do" in a similar sense to "make do" at least as early as 1851. From T.S. Arthur, "Time, Faith, Energy," in the [Sumterville, South Carolina] Sumter Banner (May 28, 1851):
And from a letter from Mrs. Judson to Mr. Fletcher dated December 14, 1853, reprinted in the [New York] Daily Tribune (March 1, 1855):
Neither of these instances is exactly equivalent to "make do" in its idiomatic sense, but both seem to express something similar—"make it suffice" rather than "make [to] suffice." Still I think that the two expressions may be closely related. The biggest challenge (and one that I can't adequately answer) is to explain how, in the 1849 instance, when the writer deemed "make do" odd enough to justify putting quotation marks around it, it already exhibited the fully formed "make do with X" construction that it commonly possesses to this day.