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The phrase back of a (or the) napkin is used figuratively to indicate that something (such as a calculation, a plan, etc.) has been formulated or devised hurriedly, roughly or carelessly (as though sketched or scribbled on the back of a napkin), and is lacking in detail or not fully formed.
—Synonyms: back of a cigarette packet and back of an envelope.

The actual practice of jotting down something on the back of a napkin was evoked, for example, in the following two texts:

1-: From the column Dayman’s Diary, published in The Stage and Television Today (London, England) of Thursday 10^th April 1986 [page 25, column 5]—the following happened during a soirée in the floating club Hispaniola, held for potential investors in the Playhouse Theatre Company:

I always thought the days of the big spenders in theatre were over. Not so. At the Playhouse do, I actually witnessed one city fat cat scrawling furiously on the back of a napkin. “Present that at my bank, my boy, and they will pay up,” he gushed. And sure enough, on closer inspection it was a hastily created, custom designed cheque.
I’ve heard about “laughing all the way to the bank” but that little piece of pure theatricality was ridiculous.

2-: From Perth’s cafe society plans rail revolution, published in The Scotsman (Edinburgh, Midlothian, Scotland) of Thursday 1^st August 1996 [page 23, column 6]—the Scottish businessman Brian Souter (born 1954) is the chairman of Stagecoach Group:

Never underestimate understatement. Over a frugal lunch of tuna sandwich and diet cola in the corner cafe, Mr Souter’s ambitions for one of the country’s fastest growing companies are boundless. As the gossiping matrons of Perth sip their tea at the next table, Brian Souter is happy to sketch out Stagecoach’s future on the back of a paper napkin.
[…]
[…] While regulators are mouthing concern and promising scrutiny, Perth’s back-of-a-napkin accountant has all the answers ready.

The earliest attributive uses of back of a (or the) napkin that I have found are as follows, in chronological order—this phrase seems to have originated in American English:

1-: From Tourist Destination Development, in The Tourist Business (Chicago: Institutions/Volume Feeding Management Magazine, 1972), by Donald E. Lundberg [3^rd printing (May 1973), page 150]:

Whereas formerly decisions to pursue a modest real estate investment were made by “back of napkin” calculations, today’s destination resort developments, involving $200 to $300 million gross investments, require careful and systematic planning.

2-: From Seaboard okays talks on track purchase by county, by Patrick McMahon, published in the St. Petersburg Times (St. Petersburg, Florida, USA) of Tuesday 25^th November 1975 [page 3B, column 4]—Merrett R. Stierheim was a member of Pinellas County’s delegation discussing with the Seaboard Coast Line Railroad the possible purchase of the railroad’s right-of-way from Belleair south to the Seminole area:

Stierheim, after the meeting, in a back-of-a-napkin quick estimate suggested that the Seminole to Belleair section could cost $5-million to $10-million.

3-: From Auto-body firm: From preposterous to profit, by Robert J. Hagen, about a company called Bradley Automotive, published in the Minneapolis Tribune (Minneapolis, Minnesota, USA) of Sunday 6^th March 1977 [page 13C, column 3]:

The company’s relatively new prosperity means it could sink something close to $1 million into developing the Bradley GT II, compared with the back-of-the-napkin brainstorming and $2,000 in materials that went into their first car, the Bradley GT.

4-: From an article about the Canadian politician Brian Smith (born 1934), a Socred (i.e., a member of the British Columbia Social Credit Party), published in The Vancouver Sun (Vancouver, British Columbia, Canada) of Tuesday 8^th July 1986 [page B2, column 3]:

In the late ’70s, Smith teamed up with the late Hugh Harris, then executive-director of the party, in an effort to modernize the party.
The two looked at political organizations across Canada and the U.S. to serve as models. In the process, Harris persuaded members of Ontario’s Big Blue Machine to come to B.C. and help turn the Socreds from a back-of-the-napkin party into a modern machine.

5-: From Arizona State discovers high price of Cardinals, by Bob Hurt, published in The Arizona Republic (Phoenix, Arizona, USA) of Thursday 15^th September 1988 [page E6, column 2]—the U.S. businessman Bill Bidwill (1931-2019) was the owner of the Arizona Cardinals of the National Football League (NFL):

As a rule of thumb, visitors figure on getting 38 percent of the gross. Ballpark figures: Multiply $38 average ticket by attendance of 70,000. Total: $2.66 million, which multiplied by 38 percent for the visitors’ take makes the trip worth $1,010,800 for Dallas.
Bidwill claims the $38 widely used as his average ticket price is incorrect. My back-of-a-napkin type figuring, based on a sellout, is a $37.80 average—before taxes.
Either way, Bill won’t have to stand in the food-stamp line.

The following are two early attributive uses of variants with cocktail napkin:

1-: From The Spokesman-Review (Spokane, Washington, USA) of Wednesday 25^th April 1984 [page 11, column 1]:

EUGENE, Ore. (AP)—One of the major differences between high-tech big business and low-tech small business, Ray Ennis will tell you, relates to start-up time.
He says complex companies seem to emerge quickly from cocktail napkin figuring to places on the New York Stock Exchange.
But Ennis’ Companion Carrier, a bike seat he designed and manufactured for small children, is now emerging after nine years of research, development and marketing studies.

2-: From the column House Works, by Steve Batie, published in the Lincoln Journal (Lincoln, Nebraska, USA) of Friday 21^st April 1989 [page 6X, column 4]:

I GUESS the worst design flaw of all is in the kitchen.
One of the things I really wanted in my kitchen was a butcher block counter with a knife rack. […]
[…]
I’m talking a serious knife rack here.
So I drew it into every kitchen remodeling plan I considered—including the final one.
I built it right into the end of the butcher block counter just where I was supposed to, and it came out just like I’d drawn it. Beautiful.
And flawed, flawed, flawed.
In fact, it’s the perfect illustration of a design flaw, because the problem was drawn in on that first little back-of-a-cocktail-napkin sketch.
The rack’s on the wrong end of the counter.