Early end to the Vietnam war?
I'm thinking that if the Vietnam war, or US participation in it, had ended earlier that might have helped some. Another option might have been keeping US involvement more low-key. Maybe something that kept South Vietnam from falling apart as much as it did in the welter of coups and counter-coups of late 1963/1964.
Here's another semi-related possibility: Nixon wins in 1960. The Soviets kick our butt in 1960-64 with a bunch of space spectaculars. Kennedy runs again in 1964, with a moonshot as the centerpiece of his campaign. Another option: Kennedy avoids assassination, is a two-term president, and is followed by 8 years of Johnson starting in 1968, assuming Johnson isn't too old by then. Of course none of this works without a less divisive and costly Vietnam war.
Of course none of that would eliminate the main obstacles to a Mars shot: (1) Cost, (2) Lack of assured return beyond prestige, (3) Risk of losing astronauts to a solar flare or some random accident, and (4) Length of the trip, with all of impacts of that much micro-gravity on the human body and all of that alone time on the human psyche.
Maybe a breakthrough in propulsion? Some kind of fusion? Of course we don't even have practical fusion power 25 years later, so it's hard to imagine it happening in time for a Mars shot in 1985. Fission-powered rockets might do the job with enough development time and political support, but I'm not sure how much support there would be for sending a bunch of nuclear material up there. Maybe if the anti-nuclear movement hadn't gotten so strong. But what would have prevented that?
Hmmm. So far I've been assuming US expeditions, but a Soviet attempt might have been more likely. Let's say that the Soviets get their super-booster working in time to beat us to the moon, or at least overshadow us by doing something we didn't--maybe a larger, more sophisticated mission, with what looks like the beginnings of a permanent base. That could work. Let's say that they beat us in sending a manned craft around the moon. We land first. They then somehow top us in terms of the landing within a week or two. The two powers then continue the space race, and the next objective for both sides is Mars. The moon and Mars missions act as a technology driver for both sides, and we develop fission or fusion-powered rockets, or some other technology. Assuming that the scientific breakthroughs are there to be made, that might give us a shot at making them.
Have to figure out some way of avoiding or downgrading the Vietnam war though. Even with a continued race with the Soviets, the mood in the US would turn against a wide range of technologies if the war continued.