According to the Oxford English Dictionary (2^nd edition, 2009), the original meaning of the phrase dead-cat bounce¹ is, in stock-market slang, a rapid but short-lived recovery in prices after a sharp fall; this dictionary also states that the phrase is of American-English origin and the earliest quotation that it provides is from 1985.

¹ The words dead and cat must be hyphenated as a compound adjective modifying bounce.

However, the earliest occurrence of dead-cat bounce that I have found is British English and denotes a rapid fall in the stock market with hardly any reaction; it is from Are City chickens coming home to roost—or is it just a dead cat bouncing?, by Victor Keegan, published in the financial section of The Guardian (London and Manchester) of Tuesday 22^nd September 1981; the following is the beginning of this article:

Fifty years ago yesterday the pound fell by more than a quarter on the foreign exchange markets against the dollar. The suspension of the gold standard (which caused the fall) and the ill-fated attempts of the Snowden Budget to balance the books have ensured that September 1931 has gone down as the watershed of English history between the wars.
Whether September, 1981 is a similar turning point is, of course, much more difficult to judge. In a sense what happened yesterday—or, rather, what didn’t happen yesterday—was quite ominous. Normally, after such a resounding drop in the share markets as happened last Friday you would have expected some form of technical rally.
In fact the market continued its free fall with hardly any reaction at all. This is known in the trade as a “dead cat bounce.” Instead, the Financial Times index of ordinary shares dropped 10 more points, though it did turn up a bit just before the close of dealings to finish 8.5 points down on the day.

However, the subsequent occurrences of dead-cat bounce that I have found have the meaning indicated by the Oxford English Dictionary, as in the following from Oil stocks no bargain, experts say, published in The Pittsburgh Press (Pittsburgh, Pennsylvania) of Monday 28^th April 1986; this article also explains that the phrase is based on the notion that even a dead cat will bounce slightly if dropped from a sufficient height:

One of the most vivid, if a bit indelicate, word pictures painted by the bears on oil comes from Raymond F. DeVoe Jr., at the investment firm of Legg Mason Wood Walker. DeVoe suggests the printing of a bumper sticker reading: “Beware the Dead Cat Bounce.”
“This applies to stocks or commodities that have gone into free-fall descent and then rallied briefly,” he said.
“If you threw a dead cat off a 50-story building, it might bounce when it hit the sidewalk. But don’t confuse that bounce with renewed life. It is still a dead cat.
“The spot oil price has recovered from under $10 a barrel to over $13 – but that also should not be confused with renewed life.”

On Tuesday 20^th October 1987, The Philadelphia Inquirer (Philadelphia, Pennsylvania) reported on “the most brutal stock-market correction in history” that had taken place the previous day; the following shows how speculators buy shares when prices are low in order to quickly resell them during the dead-cat bounce:

Jay Lockman, a 25-year veteran Philadelphia commodities trader, said he stopped by at the close to watch financial history in the making.
“Today, I’m far happier to be in commodities and not equities,” he said.
But when the public panic is over, Lockman said, he probably will buy stocks in anticipation of a “dead-cat bounce” on Wall Street.
“They say that even a dead cat bounces if you drop him off a tall building,” he said. “Maybe stocks will be my dead cat.”

The phrase has come to be used in the generic sense of a brief, spurious improvement. However, it is impossible to determine whether the earliest instance of this generic use that I have found refers to the stock-market term or is an independent coinage; this instance is from an article published in several Texas newspapers on Wednesday 2^nd September 1992, for example in The Galveston Daily News, which detailed how Ann Richards (1933-2006), Democratic Governor of Texas, “spent her birthday Tuesday firing verbal barbs at President Bush” (George Bush (born 1924), Republican President of the USA from 1989 to 1993):

Richards, with incumbent troupe in tow, blamed Bush and former President Ronald Reagan for “destroying” Texas oil and gas industry. She also predicted that the tone at this year’s GOP convention would turn away voters in November.
“The Republicans had such a really such sad convention, they didn’t get the bounce from it, called a dead cat bounce,” she told about 200 people, mostly hunters. “You know even a dead cat will bounce if you throw it hard enough.”

The second-earliest instance of this generic use that I have found clearly establishes its relation with the stock-market term; it is from Survival in the Brighton crucible, an article about the Conservative Party annual conference at Brighton, published in The Guardian (London and Manchester) of Tuesday 6^th October 1992:

John Major²’s administration is a mere six months old, yet already everything’s sagging. The pound took another terrible battering yesterday, with its only advertised hope of recovery consisting of what exchange dealers call the “dead cat bounce” — the point where a currency has fallen so far that there has to be some reaction. […]
The principle of dead cat bounce may apply to party morale as well as to the pound. With expectations so low, even a sniff of hope will look like revival.

² John Major (born 1943), British Conservative statesman, Prime Minister 1990-97