His ploy to end the Vietnam War early in his term has to actually work. He used the 'madman' theory, suggesting he was willing to start all out global nuclear war to avoid losing in Vietnam. The USSR figured he was serious, the North Vietnamese didn't. (Or rather: their grasp of geopolitics seems to have been so limited that they simply didn't understand the implications.)
If the USSR gets scared enough, and makes very clear to North Vietnam what the risk is (and that if it happens, the USSR will not back them), then Nixon can get a negotiated peace in 1969. Vietnam will be divided like Korea, and Nixon can call it "peace with honour". The anti-war counterculture evaporates in the face of no more war. The American servicemen can say that their many sacrifices meant something.
Nixon ends up the unifying figure who healed a terrible wound.
We may be reasonably certain that these ATL outcomes would prevent the whole Watergate thing later on, because Nixon would be in a very different position by then.
For bonus points:
1) Appoint conservative Supreme Justices to appease the GOP's right wing. This means that there are conservative Justices taking the place of Burger, Blackmun and Powell. (These three conservative Justices, together with White and Rehnquist, will have a majority to make Roe v. Wade go the other way.)
2) In return for a very conservative Court, the conservative wing will have to agree to no War on Drugs, and addiction to be treated purely as a medical problem, rather than a fundamentally criminal one. That solves a lot of future problems, keeps the future drug cartels from essentially getting state protectionism, and underscores Nixon's reputation as "healing the nation".
3) Any other VP than Spiro Agnew, please.
(Edited now.)
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