Oddly enough, it was Bush and a crew of two other Nixon administration members that floated on the periphery that would come to have a heavy hand in Republican administrations for the next three decades or so--Donald Rumsfeld went on to serve as Chief of Staff and twice as Defense Secretary, while Dick Cheney became the House Minority Whip, Secretary of Defense, and Vice President. The first Republican Governor of Texas since Reconstruction, Bill Clements, served as Deputy Secretary of Defense for Nixon and Ford.
In general, not many people usually run for office after membership in a presidential cabinet (outside of the Vice Presidency). Aside from Hillary Clinton and George H.W. Bush, how many non-Vice Presidential members of Presidential administrations since... 1960 have been on a ticket? Sargent Shriver, Dick Cheney... I wouldn't attribute the rise of Bush, Rumsfeld, and Cheney to some overall trend rather than to serendipity. If you are overly deterministic, you could say that Bush's victory in Iowa gave him the presidency, which gave Cheney the Vice Presidency, which made Rumsfeld Secretary of Defense, but that was largely dumb luck. John Tower was supposed to be George H.W. Bush's Defense Secretary, which would have eliminated Cheney's time in the cabinet, and George W. Bush, I believe, originally wanted the CEO of Federal Express to have the Pentagon job (and I can imagine Tom Ridge definitely being on the table as well). I think it just so happens that these three men were particularly canny and fortuitous during a time when the Republicans were more often in the White House than out of it. A President's son seeking the office is unique in its own right.
Andrew Cuomo, Bill Richardson, Lee P. Brown, Deval Patrick, and others might beg to differ with the bolded words.
