The office of Vice President of the United States is often a lonely one, without many official responsibilities. A joke became popular on 1896:
 
“I believe he had two sons; one of the them was lost at sea, and the other became vice president of the United States.”
“And, of course, neither of them was heard of afterward?”—Truth.
 
New York-born Theodore Roosevelt (1858-1919) told the joke in 1900, before he became vice president in 1901. Many other vice presidents have repeated the joke.
 
   
Wikipedia: Thomas R. Marshall
Thomas Riley Marshall (March 14, 1854 – June 1, 1925) was an American Democratic politician who served as the 28th Vice President of the United States (1913–21) under Woodrow Wilson. A prominent lawyer in Indiana, he became an active and well known member of the Indiana Democratic Party by stumping across the state for other candidates and organizing party rallies that later helped him win election as the 27th Governor of Indiana. In office, he proposed a controversial and progressive state constitution and pressed for other progressive era reforms. The Republican minority used the state courts to block the attempt to change the constitution.
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One of his favorite jokes was about a woman with two sons, one of whom went to sea and one of whom was elected vice president; neither was ever heard of again.
         
4 August 1896, Logansport (IN) Pharos, “humor,” pg. 4, col. 4:
The same fate.
“I believe he had two sons; one of the them was lost at sea, and the other became vice president of the United States.”
 
“And, of course, neither of them was heard of afterward?”—Truth.
 
16 August 1896, San Francisco (CA) Chronicle, “Humorous Side of Life,” pg. 11, col. 2:
“I believe he had two sons; one of the them was lost at sea and the other became Vice-President of the United States.” “And, of course, neither of them was heard of afterward?”—Truth.
 
Google Books
29 August 1896, The Outlook, “Bits of Fun,” pg. 402, col. 2:
“I believe he had two sons; one of the them was lost at sea and the other became Vice-President of the United States.” “And, of course, neither of them was heard of afterward?”—Truth.
 
Old Fulton NY Post Cards
14 February 1900, New York (NY) Press, “Whence No Man Ever Returns,” pg. 6, col. 6:
When a powerful emissary of the gang which wishes to unload Governor Roosevelt was working on him to get out of the race for Governor, Teddy told him this story: “Once there was a man. he had two sons. One of them went to sea; one became Vice President. neither was ever heard of again.” Teddy is dead on to Mr., the Hon., Senator T. C. P.
     
22 February 1900, The Daily Nonpareil (Council Bluffs, IA), pg. 7, col. 4:
This is said to have been Governor Roosevelt’s reply to a politician who suggested to him that he accept second place on the republican ticket: “Once there was a man who had two sons. One of them went to sea, the other became vice president and neither was ever heard of again.”
 
March 1904, United Service; a Quarterly Review of Military and Naval Affairs, “President Roosevelt” by Sydney Brooks, pg. 286:
THE Vice-Presidential office is one of the least satisfactory features of the American Constitution.
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There was truth as well as acrid humor in the parable of the American who had two sons. One of them went to sea, the other became Vice-President. Neither was ever heard of afterwards.
   
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Front Runner, Dark Horse
By Ralph G Martin and Ed Plaut
Garden City, NY: Doubleday
1960
Pg. 81:
... DiSalle opened with the story of the man who had two sons: one who went to sea and the other who became Vice-President — and neither was ever heard of again.
 
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Kennedy:
A Time Remembered
By Jacques Lowe
New York, NY: Quartet Visual Arts Book
1983
Pg. 96:
Hubert Humphrey would say ruefully: “There is a story about the mother who had two sons. One went to sea and the other became vice president, and neither was ever heard of again.”
   
Google Books
Race to the Moon:
America’s Duel with the Soviets
By William B. Breuer
Westport, CT: Praeger
1993
Pg. 160:
Johnson was fond of quoting the old cliche: “A mother had two sons. One went to sea, and the other became vice president of the United States, and neither was ever heard from again.”
 
Google Books
The American Vice Presidency Reconsidered
By Jody C. Baumgartner
Westport, CT: Praeger Publishers
2006
Pg. 135:
Harry Truman’s vice president, Alben Barkley, liked the joke about a man who had two sons: One ran away and went to sea, the other was elected Vice President of the United States. Neither was ever heard from again.
   
New York (NY) Times
Speak Up, Mr. Vice Presidential Candidate. We Can Barely Hear You.
The Conversation
By ARTHUR C. BROOKS and GAIL COLLINS OCT. 4, 2016
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Gail: My favorite vice president was Thomas Marshall, who had two terms of being totally ignored by Woodrow Wilson. Even when Wilson was half-dead from a stroke, Marshall wasn’t consulted. He used to tell a story about a woman who had two sons: One ran off to sea and one became vice president of the United States. Neither was ever heard from again.