Could he have done something to diminish or perhaps prevent the economic crisis that occurred during the last years of his presidency, a better decision in this or that area? That would do wonders to reduce political unrest.
Some of the things that have to change for Perón to
not face a coup (his actual deposition in said coup is a separate issue) are too intrinsically tied to both his personal beliefs and his party's ideology; his closure of
La Prensa for example isn't something that can be changed without fundamentally altering what that early peronism was, which included an opposition to critical press and a hostility to political opposition in general. The same is true of his economic policies: the austerity of his second term was a consequence of his profligacy in his first term for instance, and changing those policies requires a POD in 1946.
Here's another thing I just found: Arturo Illia, who IOTL was president from 1963 to 1966,
ran for governor of Córdoba in 1951 and lost by around 9 points, a relatively narrow margin considering the national Peronist landslide. What could've happened if he won? Could the opposition (or at least the UCR) rally to him as a potential presidential candidate instead of agitating for and planning a coup?
Perón was deposed because of military opposition, his civilian opposition was already thoroughly neutered and reduced to desperate resistance as their leaders were jailed, murdered or exiled. If Perón's overthrow is prevented, the UCR candidate in 1957 would likely have been Balbín, who was the
de facto leader of the opposition already.
At any rate, I can think of two ways to prevent Perón's overthrow: there was a military insurrection earlier in the year which failed, and Perón could purge the general staff in response which would make the second insurrection more difficult and less likely to occur; alternatively, he could have refused to resign and fought back, which would likely have lead to a civil war - and it's a shame that Maverick's TL (La Larga y Oscura Noche) was purged from the site when he decided to leave the forum, since it covered this very topic and was one of the best (and for a while one of the few) South American timelines here.
There's actually a good chance that Perón or a candidate of his choosing would lose in 1957: his popularity would continue to decline as the economy worsens, and without the 1955 coup, the 1956 split in the UCR (which would lead to Frondizi's creation of the UCR-I and his subsequent endorsement by Perón) might be butterflied away. Balbín-Frondizi is the likely opposition ticket in the next election.