Alright, how about this.
In 1960 Nixon decides to go with Everett Dirksen for his Vice President instead of Henry Cabot Lodge. With Dirksen on the ticket Nixon carries Illinois. Also Dirksen doesn't make any statement about putting an African-American in the Cabinet, which leads to Nixon doing slightly better in the south, so Nixon also carries Texas and with it the election.
President Nixon offers the Secretary of the State position to Nelson Rockefeller. Rockefeller hems and haws a great deal about whether it is worth giving up being the first man in New York for a subordinate role in D.C., but ultimately the prospect of being a major player on the world stage is just to great for him, and he accepts the Secretary of State position. Nixon goes on to win reelection in 1964 on the strength of a good economy and the prestige from having overthrown Castro.
The Nixon Administration achieves a number of foreign policy successes in its 2 terms. Overthrowing Castro in the first term and opening up relations with China and achieving arms control accords with the Soviets in the second term. (South Vietnam is still a mess, but since Nixon doesn't authorize the overthrow of President Diem, the South Vietnamese government is able to stagger along without the large scale deployment of US troops to the country.) Rockefeller as the Secretary of State is largely associated with these foreign policy successes (even though President Nixon actually kept Secretary of State Rockefeller largely sidelined from his major foreign policy initiatives and mainly had Rockefeller dealing with NATO, Africa and Latin America while Nixon handled the Soviets and Chinese), and thus Rocky is able to use the prestige from his successful tenure as Secretary of State (along with the support of President Nixon who is supporting Rockefeller over his main opponent for the nomination, Senator Goldwater, since Goldwater has been highly critical of Nixon opening up relations with China and working for detente with the Soviets). Thus Rockefeller ends up winning the 1968 Republican presidential nomination.
As for RFK, since his brother was not elected in 1960, he doesn't have a Cabinet post. RFK therefore runs for and wins Governor of Massachusetts in 1962 (and gets reelected in 1964). JFK decides not to run for president in 1964 (his health is not great, so he doesn't really want to go through another presidential campaign), so an unenthusiastic Democrat Party nominates LBJ that year who goes on to lose to Nixon in the fall. Meanwhile RFK is proving a successful Governor. He earns a reputation for being tough on crime (enacting some of the toughest anti-drug laws in the country) while also a champion for civil rights after he secures the enactment of open housing legislation in Massachusetts. With his brother again deciding not to run for the presidency in 1968, the path is open for RFK, and with the rising national crime rate and civil rights proving to be major issues that year, RFK is able to use his record as Governor of Massachusetts to secure the Democratic nomination.
As for running mates:
I could see Rockefeller considering Senator Thurston Morton of Kentucky, Governor Jim Rhodes of Ohio, Governor George Romney of Michigan, Congressman Gerald Ford of Michigan and (assuming they won their 1964 elections which IOTL they all narrowly lost) Senator Howard Baker of Tennessee, Senator Robert Taft of Ohio, Senator Paul Laxalt of Nevada and Governor Charles Percy of Illinois.
As for Kennedy, he's almost certainly facing a 3rd party challenge from George Wallace, so he probably needs a moderate southerner. Maybe former Governor Terry Sanford of North Carolina. Senator Al Gore of Tennessee could also be a possibility if any civil rights legislation pushed by President Nixon was sufficiently watered down that Gore was able to vote for it.
I imagine even with a Wallace 3rd party candidacy that Kennedy would win. The Republicans will have been in power by 16 years at this point, so the country will be ready to give the Democrats a chance, and the right wing of the Republicans is going to be very unenthusiastic about a Rockefeller candidacy.
I don't really know enough about RFK to speculate that much on how his presidency is likely to go. However, based on things that RB has said in the past, it sounds as though IOTL RFK became much more compassionate and concerned for the plight of the less fortunate after his brother's murder. ITTL he hasn't gone through such a wrenching. life changing experience, so President Robert Kennedy here might be much more about fighting a war on crime rather than a war on poverty. He probably will still aggressively pursue desegregation in the south though.