Money (such as coins and a wallet) is often kept in the pockets on a person’s garment. A “deep pocket” would be able to hold more money. Lawyers developed legal strategies to go after individuals or companies with “deep pockets.”
 
The term “deep pockets” (also given as “deep pocket” and “deep pocketed”) is attested sparsely in the 1940s through the 1960s, but became popular with the litigation explosion of the 1970s.
 
A person with “short arms” and “deep pockets” is a person (sometimes derided as “miserly” or “cheap”) who saves money and doesn’t often spend it. The term “short arms and deep/long pockets” is cited in print from at least 1946, 1950 and 1952.
     
 
Wikipedia: Deep pocket
Deep pocket as a slang term
Deep pocket is an American slang term; it usually means “extensive financial wealth or resources”. It is usually used in reference to big companies or organizations (ex: the American tobacco companies have “deep pockets”), although it can be used in reference to individuals (e.g., Bill Gates, Donald Trump).
 
In the context of a lawsuit, the deep pocket is often the target defendant, even when the true (moral) culpability is with another party because the ,i>deep pocket has money to pay a verdict. For example, a lawyer may comment that he or she sued the manufacturer of a product rather than the seller because it is the deep pocket, meaning it has more money than the seller with which to compensate the victim.
 
Deep pocket in law and economics
Deep pocket is a concept often used in the law and economics of tort law. It refers to the idea that the risk of an activity should be borne by a person that is in a relatively good position to handle it. This can be achieved by either spreading the risk over a large number of risk-bearers (usually by means of insurance), or by imposing it on a person that is relatively risk-neutral. The latter is often assumed to be the case for wealthy individuals or large corporations, who are referred to as having “deep pockets”, since their wealth will not be affected very strongly if the risk materializes. For example, a deep pocket argument might, among other arguments, be used to justify product liability, as producers with “deep pockets” will normally be better able to accommodate the risk of damages than individual consumers not endowed with “deep pockets”.
 
UsingEnglish.com
Idiom Definitions for ‘Deep pockets but short arms’
Someone who has money but never puts his hand in his pocket to pay for anything has deep pockets but short arms.
 
Google Books
The New Partridge Dictionary of Slang and Unconventional English
By Tom Dalzell and Terry Victor
New York, NY: Routledge
2006
Pg. 1736
have short arms and long pockets
to be stingy AUSTRALIA 1966
   
(Historical Dictionary of American Slang)
deep pockets n. pl. an inexhaustible source of revenues, abundance of money. Also sing.
1976 Business Week (Nov. 1) 64: Loews was a new, deep pocket.
1976 (cited in W10).
1977 Wash. Post, in Barnhart Dict. Comp. V 11: He knows all his alumni with the big bucks and those deep pockets are watchin’ and they’d be mighty upset to lose to us.
1984 U.S. News & W.R. (Dec. 3) 63: For those with deep pockets, Neiman-Marcus has a handmade, wooden desk…for $65,000.
1993 N.Y.Times (June 23) III 1: Boeing officials say they are competing not just against a company—but against the deep pockets of the French, German, British and Spanish governments.
       
Merriam-Webster Dictionary
Main Entry: deep pocket
Function: noun
Date: 1975
1: a person or an organization having substantial financial resources
2plural : substantial financial resources < a corporation with deep pockets >

— deep–pock·et·ed \ˈdēp-ˈpä-kə-təd\ adjective  
   
6 August 1895, Atlanta (GA) Constitution, pg. 6, col. 1:
In short, the money sharks not only have deep pockets but long arms, and they have in their employ some of the brightest newspaper writers of the east.
   
Google Books
Oil Across the World: The American Saga of Pipelines
By Charles Morrow Wilson
New York, NY: Longmans, Green and Co.
1946
... Pennsylvania, and abetted by the burly Cap’n Jake Vandergrift, of the fabulously deep pockets and the oil-stained folding money, set about building an oil refinery at Oil City, Pennsylvania, and a pipeline to carry the finished products to Wilkes-Barre for rail shipment via the Jersey Central to New York.
 
14 September 1946, Milwaukee (WI) Sentinel, Lloyd Larson column, pt. 2, pg. 4, col. 1:
(Spoken by Junior Kibitzer.—ed.)
“The moderate wealthy and the comfortably usually get that way by being sound businessmen—and along the line they get into the habit of tossing the ‘lettuce leaves’ (dollars—ed.) around like pianos. Short arms and deep pockets!”
         
5 October 1946, Milwaukee (WI) Sentinel, Lloyd Larson column, pt. 2, pg. 3, col. 7:
Neither Short Arms Nor Deep Pockets
(Spoken by Junior Kibitzer.—ed.)
“BRANCH RICKEY always had a reputation as one of the smartest operators in baseball, but no one—at least I never heard of him—ever gave him credit for being big hearted until he started shelling out to the hired hands. First he gives each Brooklyn (Dodgers—ed.) player a new car.”
 
17 November 1950, Windsor (ON) Daily Star, pg. 12, col. 6:
She Knew Him
BARNET, Hertfordshire, Eng. (CP)—A woman giving evidence in a local court case said: “My husband isn’t really mean. He’s just got short arms and deep pockets.”
 
8 February 1952, The Advertiser-Gleam (Guntersville, AL), pg. 3, col. 4:
“He sure is stingy.” “Oh, don’t be so hard on him. He’s just got long pockets and short arms.”
   
1 April 1952, Orlando (FL) Evening Star, “Around the Town” by Hanley Pogue, pg. 6, col. 7:
He’s not really stingy. He just has long pockets and short arms.
 
10 December 1952, Danville (VA) Bee, “The Beehive,” pg. 6, col. 6:
He’s not really stingy. He just has long pockets and short arms.
     
6 June 1953, Indiana (PA) Evening Gazette, “Teen Town Talk” by Florence Faus, pg. 3, col. 3:
...Deep-Pocket Lads—fathers;...
 
9 December 1953, Hartford (CT) Courant, “Courant Camera Quiz” by Arman J. Hatsian, pg. 11, col. 3:
(Answered by Ortiz Walton.—ed.)
The only way to save money at Christmas time is to have short arms and long low pockets.
       
20 June 1957, Philadelphia (PA) Inquirer, “Cubs’ Pilot Sees Phils In There, However” by Don Daniels, pg. 24, col. 3:
YOU don’t have to have an abacus to figure out that the National League race is tighter than a loan shark with short arms and long pockets.
 
Google Books
From a Trade to a Profession:
Byways in Dental History
By John Menzies Campbell
Published by The Author
1958
Pg. 9:
I purposely use the word lure because a few seceders, having proved that a policy of ‘short arms and deep pockets’ failed to satisfy, later returned.
       
Google Books
The Reporter
Passaic County Bar Association
1959
Pg. ?:
Short arms and long pockets with their own dough, but liberal with yours.
 
16 June 1959, Montana Standard (Butte, MT), pg. 15, col. 3 classified ad:
ONE BEAUTIFUL, BEAUTIFUL BOSTON female, very reasonable. 3 Boston male puppies, on the puppies please don’t have short arms and deep pockets. Copperhill Kennels.
 
Time magazine
The Biggest Fund
Monday, Dec. 21, 1959
For the past three years, Harvard University has diligently appealed to its alumni in a well-planned drive to raise $82.5 million. Purpose: a lavish refurbishing of Harvard College (TIME, Nov. 26, 1956). Last month, still about $10 million short of the goal, Harvard went back to wealthy alumni who had already given. Last week the results were announced: out of deep pockets in three weeks flowed 18 six-figure gifts totaling $3,100,000, to boost the pledges to $75 million.
 
Time magazine
Somebody Got Him
Friday, Aug. 07, 1964
(...)
Hoffa huffed that he would pay “out of my own pocket,” but that takes some mighty deep pockets, even with his $75,000-a-year salary and his other well stocked resourses. Transcripts of the Chicago testimony alone may cost him $19,500.
   
1 September 1965, New York (NY) Times, “Aspen Debates Its Image: Devotees of ‘Deep Pocket’ Tourists Fear Creation of a ‘Hot Dog’ Circuit” by Grace Glueck, pg. 31:
“Aspen has been catering to the ‘deep pocket’ tourist and if the road is paved it might be more or less on a ‘hot dog’ circuit leading to an influx of an entirely different economic group,” the Mayor, a rancher, vice president of the Aspen Institute for Humanistic Studies and a scion of Milwaukee’s Pabst brewikng family, is reported to have said.
     
Google Books
Securities and Federal Corporate Law
By Harold S. Bloomenthal and Samuel Wolff
Published by Clark Boardman Co.
1972
Pg. ?:
Entrepreneurial lawyers also like to sue deep-pocketed professional advisers, like accountants and lawyers, even if they are only marginally involved in the alleged fraud.
   
Google Books 
Sue the B*st*rds:
The Victim’s Handbook
By Douglas Matthews
Published by Arbor House
1973
Pg. 134:
...defendants have what lawyers lovingly refer to as deep pockets—ability to pay—and there is less tendency to sympathize with their vulnerability to larger judgments.
     
Google Books
Running Your Own Business:
A Handbook of Facts and Information for the Small Businessman
By Howard H. Stern
Pasadena, CA: W. Ritchie Press
1976
Pg. 5:
Lawyers always go after the guy with money. This is called the deep pockets principle.
     
Google Books
Health: A Victim Or Cause of Inflation?
By Michael Zubkoff
Milbank Memorial Fund
Edition: 2
Published by Taylor & Francis
1976
Pg. 276:
Relying for payment on the deep pockets of third-party payers, institutions, too, have accepted the invitation extended by insurance to expand their services without regard to cost.
     
Google Books
The Equity Funding Papers: The Anatomy of a Fraud
By Lee J. Seidler, Frederick Andrews and Marc J. Epstein
Hoboken, NJ: Wiley
1977
Pg. 6:
In legal parlance, they have “deep pockets,” eminently deep pockets.
     
Time magazine
Money, Money, Money
Monday, Nov. 20, 1978
(...)
While candidates across the South repeatedly denounced high government spending, they were less critical of campaign spending. The old Confederacy was awash with money, much of it from the candidates’ own deep pockets.
 
Google Books
Frommer’s Scotland and Wales on $35.00 a day
By Darwin Porter
New York, NY: Frommer/Parmantier
1986
Pg. 333:
The town fathers were said to have “long pockets and short arms.”