The only two times it actually happened were John Adams/John Quincy Adams and George H. W. Bush/George W. Bush. (Benjamin Harrison was William Henry Harrison's grandson, not son.)
Some instances it could have happened:
(1) Had John Van Buren
https://en.wikipedia.org/wiki/John_Van_Buren enjoyed better health and been true to his early antislavery principles, he would certainly have been enormously influential in the newly formed Republican Party. He might well have become their presidential candidate in 1856 or 1860--or he might at least have become vice-presidential candidate in 1860, and if renominated in 1864 become president after Lincoln's death. Henry Wilson, radical Republican and vice-president under Grant (who as an ex-Whig had no special leanings toward ex-Democrats) wrote in his *History of the Rise and Fall of the Slave Power in America, Volume 2*, p. 142:
"Indeed, such was the brilliant record he then [1848] made, his popular talents, his prestige of name and position, that, had he remained true to the principles he then advocated, he would unquestionably have been one of the foremost men of the Republican party, if not its accepted leader..."
https://books.google.com/books?id=pos-AAAAYAAJ&pg=PA142
(2) Charles Francis Adams (John Quincy Adams' son) wins the Liberal Republican nomination and defeats Grant in 1872. For why I think it is unlikely he would win the nomination, see
http://groups.google.com/group/soc.history.what-if/msg/f625c076d72631a8 Even if he is nominated, it's hard for any candidate to defeat Grant in 1872, unless the Panic of 1873 and ensuing depression had started a year earlier than they did...
(3) Robert Todd Lincoln was often mentioned as a Republican presidential nominee, and in 1888 for example might well have been nominated (and would win the general election) if he showed any interest. However, in 1884 Lincoln did privately indicate that he would accept the *vice* presidential nomination under one circumstance: if Arthur were nominated for president. (Lincoln had a high opinion of Arthur; note that he was the *only* one of Garfield's cabinet that Arthur had retained.) Jason Emerson, *Giant in the Shadows: The Life of Robert T. Lincoln* (Carbondale and Edwardsville: Southern Illinois University Press 2012)* argues that if Lincoln had made this fact known to the Republican national convention, it might well have nominated Arthur, the Arthur-Lincoln ticket would go on to defeat Cleveland, and then after Arthur's death from Bright's disease, America would have its second President Lincoln. See my discussion at
https://www.alternatehistory.com/fo...coln-possibly-president.350654/#post-10604233
(4) Frederick D. Grant
https://en.wikipedia.org/wiki/Frederick_Dent_Grant narrowly lost the 1887 election for Secretary of State of New York with 452,881 votes to 469,888 for the Democratic candidate Frederick Cook.
https://en.wikipedia.org/wiki/New_York_state_election,_1887 Had Grant won, he would be a plausible Republican candidate for Governor at some time in the future, and from there he might well go on to the national ticket.
(5) James Rudolph Garfield,
https://en.wikipedia.org/wiki/James_Rudolph_Garfield son of President James Garfield, and Secretary of the Interior under Theodore Roosevelt. If Taft is for some reason unavailable in 1908, TR might want Garfield as his successor, and the GOP national convention might go along in spite of not caring much for Garfield's progressivism. (An incumbent Republican president in those days had a way of imposing his will on GOP conventions--his control of the patronage-dependent delegates from the South.)
Alternately, if TR wins the GOP nomination in 1912 (not likely, but let's say that Taft just gets tired of the presidency and decides that one term is enough for him), he might choose Garfield as his running mate. Then if the TR-Garfield ticket is elected and TR dies in office...
(6) Theodore Roosevelt, Jr. wins the governorship of New York in 1924 and becomes GOP presidential nominee in 1928. See
http://groups.google.com/group/soc.history.what-if/msg/856376cac78a4ae8
(7) Robert Taft gets the GOP presidential nomination he sought in 1940, 1948, or 1952--though I think 1952 was the only time when, had he been nominated, he would have anything like a 50/50 chance to win in November. He would certainly not win by anything like Ike's margin, but given the issues of Communism, Corruption, and Korea, he definitely had a chance to win. (And he could easily have gotten the nomination if he had only done enough to reassure Ike that he supported collective security; in that case, Ike would not have run. Their differences on *domestic* issues were not decisive; indeed, on housing, Taft was actually to the left of Ike.)
(8) There was also William Howard Taft's *other* son, Charles Phelps Taft II.
https://en.wikipedia.org/wiki/Charles_Phelps_Taft_II But though he was a highly respected Mayor of Cincinnati, the only way I can see him on a national GOP ticket is if he became Governor of Ohio. But when he ran for that office in 1952, incumbent Democratic Governor Frank Lausche was just too popular, and Taft lost by 11.79 points.
http://www.ourcampaigns.com/RaceDetail.html?RaceID=27757 Maybe if for some reason Lausche had died or chose to run for another office in 1952 (for Senator against Bricker?), Taft would have had a chance.
(9) Franklin D. Roosevelt, Jr.,
https://en.wikipedia.org/wiki/Franklin_Delano_Roosevelt_Jr. As I note at
http://groups.google.com/group/soc.history.what-if/msg/01aa04bdcf07b770 according to Steven Hess, no less a political expert than John F. Kennedy "thought that if Franklin Roosevelt, Jr., had only secured the support of Tammany boss De Sapio in 1954, he would have become the Democratic gubernatorial candidate, he would have won, and he would have been the 1960 presidential nominee." (I suppose some "birthers" would have complained about his being born at Campobello but I don't think either the courts or most voters would take that objection too seriously.)
(10) James Roosevelt (another son of FDR): He might have had a chance if not for his disastrous decision to run against the very-popular Earl Warren for Governor of California in 1950. See
http://groups.google.com/group/soc.history.what-if/msg/d2e4b1e6d0c63af9 for a scenario where Roosevelt runs for Congress instead of Governor in 1950, and eventually becomes Governor in 1958 (though IMO he would have had a hard time defeating Pat Brown in the Democratic primary) and President in 1960.
(11) If Jeb Bush had won the governorship of Florida in 1994, he rather than his brother George W. might have become Republican nominee for president in 2000. Jeb would have carried Florida in 2000 a lot more easily...
(12) Maureen Reagan *might* have gotten elected to the US House of Representatives, but I doubt she could ever have made it to the White House:
https://www.alternatehistory.com/fo...agan-how-far-could-she-go-politically.387983/
(13) Of course there's always speculation about what if JFK, Jr. had lived. "Ted Kennedy believed that politics was John's destiny, and urged him to run for several offices. Various polls between 1996 to 1999 showed Kennedy Jr as the most popular Democrat in New York. John was reportedly considering a run for
Senate in 2000 to replace retiring Senator
Daniel Patrick Moynihan. But was later said to be considered running for governor in 2002 against
George Pataki."
https://en.wikipedia.org/wiki/John_F._Kennedy_Jr. I think that by the time of his fatal crash he had already decided against running for the Senate in 2000, and if he ran against Pataki in 2002, he would almost certainly lose (and not just because it was a Republican year).
These are the main possibilities, though I am sure I have forgotten about someone or other. John Scott Harrison
https://en.wikipedia.org/wiki/John_Scott_Harrison was the son of a president and the father of another but his own political career was too brief and obscure (four years in the US House of Representatives) to give him much chance of becoming president himself IMO. Any speculation about Donald Trump, Jr., Ivanka, etc. is future history and therefore not included here. Also, I am trying to confine this list to children of *actual* presidents, rather than "X might have become president and then his son Y twenty years later."
(I chose the post-1900 section for this because the majority of my choices have POD's after 1900.)