Upon further investigation, the POD is a bit earlier than 1888 but...
This doesn't really qualify as your Challenge answered but I had been working on something in my head for a while and well... I'll need some suggestions/help with it.
Mine basically has P. J. Kennedy becoming a successful businessman earlier therefore being accepted by the Boston Brahmin. His position as perhaps the only Irish Catholic to enter the ranks of the wealthy Republicans while also being relatively populist makes him a powerful (albeit centrist) Republican leader. Joe Kennedy Sr. would go on to acquire great wealth and political influence and would become the Ambassador to Great Britain during WW2.
(While the Kennedy family's political party was the GOP, their politics were only a little less liberal than IOTL. The term "Kennedy Republican" replaces and predates "Rockefeller Republican" ITTL.)
When Secretary of Defense Richard M. Nixon, having become quite a popular statesman during the fifties, announced his desire to be considered at the convention, he allied himself with the McCarthy and Kennedy camp. However, it soon became clear that the Kennedy machine (having been campaigning long
before the primaries) was going to dominate when at the Republican National Convention the moderate-left John F. Kennedy received a tight majority. Goldwater backed out and Rockefeller immediately threw his support behind Kennedy while Nixon, (who ran a close second) was asked to become Vice President, which he somewhat begrudgingly yet gratefully accepted. The pair got along well, however, and their policies and plans for the White House were something of a partnership even down to the last detail.
From the beginning the Johnson/Humphrey ticket was trailing. Although they promised maturity and more-of-the-same Democratic policies popular in the South, by comparison they campaigned much less and appeared old. Johnson assumed his home state was safe, but thanks to Nixon's campaigns there it went to Kennedy by a narrow margin. The whirlwind campaign tour organized by Robert Kennedy was a spectacle. When not touring together, Nixon was making good on his promise to visit all 50 States during the Kennedy/Nixon tour (focusing on winning the South) while Kennedy was exciting Republicans and converting all manner of progressive and center-right Democrats (dubbed Kennedycrats by the press) with his enthusiastic and mature campaign promises and his charismatic and youthful presentation in larger states in the North.
The culmination of the campaign was a series of Presidential and Vice Presidential debates between the candidates. Kennedy absolutely nailed Johnson and in a similar debate between the VP candidates, Nixon (agreeing to prepare extensively and wear make-up at the behest of Robert Kennedy) made a more narrow but no less respectable victory over Humphrey.
In response to Kennedy's support of Martin Luther King Jr. the African American vote went almost entirely to their ticket, and both Kennedy and Nixon made a point to address the good economy and immense popularity of Eisenhower who actively campaigned for Kennedy in the final 10 days of the campaign.
The election was a landslide, wide margin win for Kennedy/Nixon.
(Long story short, Kennedy is still assassinated and Nixon replaces Johnson ITTL. Nixon wins in 1964 and 1968 largely because he agrees to allow Bobby Kennedy to be his running mate. Robert Kennedy, is still shot at in California while campaigning with Nixon though many believe Nixon was the actual target. The shooter, Sirhan Sirhan dies of wounds received during the mayhem but Robert Kennedy survives and successfully runs with Rockefeller as running mate in 1972. None of it is really fleshed out yet but a Republican Kennedy family with similar OTL politics would be an interesting thing to work on.
I'm having a hard time combining this new Nixon and this new Kennedy's policies in a backdrop of a sixties initially similar to OTL's version. What do you think would happen here? And I'm still intent on having JFK shot to death in Dallas but RFK surviving.)
Oh yeah and Maria Shriver Bush...
Prescott is on Kennedy/Nixon staff, The Bush's the Kennedy's and the Nixon's all become relatively good friends.
George W. Bush (who is much more liberal and better spoken ITTL) is re-introduced to family friend Maria Shriver in 1977 and the two strike up a relationship resulting eventually in marriage. Bush becomes governor of California before eventually running for President.
EDIT: All of this is assuming Honey Fitz still allies with a decidedly still relatively liberal P. J. Kennedy and gives their children's marriage his blessing.