AHC: An American VP voluntarily declines renomination for Vice President

Or runs for another office while serving as VP in a first term, which implies resignation as VP if the candidate wins. This does not include the case where the President does not seek re-election.

There were two such cases in the 1800s.

VP Daniel D. Tompkins (1817-1825) ran for governor of NY in early 1820. (He lost, and later that year was re-elected VP in what was the last unopposed Presidential election).

VP Levi P. Morton (1889-1893) declined to run for a second term in 1892. He was elected governor of NY in 1894.

Who in the 20th century might have done similarly?
 
I think you have to keep the Vice-Presidency "not worth a bucket of warm piss." That, and either give the VP something more meaningful to run for (Governor or Senator), or a genuine reason to want to spend more time with their family.
 
Never doubt the power of a family tragedy.

Also, you could have it play out like one of House of Cards’ less silly seasons. A Democratic Governor from the swing state of Pennsylvania resigns and the party struggles to find a good candidate. The VP is from PA and is persuaded to run and wins.

You could have it play out in any swing state and maybe even have them running for a senate seat to maintain a one seat majority or win a majority when they had previously been in the minority. Party majority’s switching mid-session are uncommon but not unheard of. As far as I remember, it happened about three times: in 2001, the ‘50s, and the 1800s. Could be fun stuff, and add an on edge level of excitement.
 
Cheney offered to drop from the ticket in 2004 due to the perception of undue influence in the administration.

If Beau Biden's health had taken a turn for the worse a few years sooner than OTL, I can imagine Joe opting out.

JFK doesn't get shot and LBJ gets sick of being VP due to it being a kind of pointless job.

Also, you could have it play out like one of House of Cards’ less silly seasons. A Democratic Governor from the swing state of Pennsylvania resigns and the party struggles to find a good candidate. The VP is from PA and is persuaded to run and wins.

Truman offered to be Ike's VP in 1948. Have an Eisenhower-Truman ticket win in 1948 and Truman decide to step down in order to run for Senate again come 1950 or 1952 (two years Missouri had races).
 
Didn't Nelson Rockefeller (Gerald Ford's VP) voluntarily remove himself from the 1976 ticket?

I don't think so:

"Ford's recollection clashed with those of Rockefeller and other administration members. Hurt by his removal, Rockefeller remarked, "I didn't take myself off the ticket, you house—he asked me to do it." He recalled Ford explaining that "it's very important that I get this nomination. And I have been talking with my political advisers and I think it would be—as much as personally I feel badly about it—it would be better if you were not on the ticket and if you would withdraw." Hartmann reluctantly concluded that in presenting Rockefeller's withdrawal as voluntary, Ford 'fudged his pledge of openness and candor with the American people.'' https://books.google.com/books?id=m2JyHO1vNOEC&pg=PA311
 
If Nixon will still have him and Agnew thinks better of it, it’s passable. It probably depends on Nixon’s temperament and how much he knows.

Agnew in OTL tried to hang on to the vice-presidency as long as he could, only resigning as part of a plea agreement. He will "choose" to leave the ticket in 1972 only if he is already in serious legal trouble--but in that case his departure could hardly be counted as voluntary.
 
Of course Ford was aware of that, too, which is why it is extremely doubtful that Rocky's withdrawal was "voluntary."

I do remember a number of liberal Republicans left with Rockefeller. They did not necessarily vote for Carter, but they did not vote for Ford. There was a feeling the party was heading away from it's Eisenhower era ideals.
 
Rockefeller was trying to convince Ford to not run in ‘76 to make room for him to be president, so I doubt he was very happy about being removed from the VP slot.

I do remember a number of liberal Republicans left with Rockefeller. They did not necessarily vote for Carter, but they did not vote for Ford. There was a feeling the party was heading away from it's Eisenhower era ideals.

Really? One of the only things I know about Ford is he got over 90% of the Republican vote and still lost because the GOP was half the size of the Democratic Party at the time. Despite his party-wide appeal, Ford couldn’t overcome Carter, who won close to half of Democrats and the presidency.
 
Agnew was considered a possible Supreme Court appointee. If he managed to survive the confirmation process, he’s out of Nixon’s way.
 
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