The novelization of Star Trek: First Contact from 1996, which is available for anyone with a scribd membership here, has a section after the story titled “A First Look at Star Trek First Contact” on p. 242 which gives some insight into the decision on this. This section describes how screenwriters Brannon Braga and Ron Moore initially decided to have a story where the Borg went back in time and threatened some part of Earth’s history, and after considering a story set in the Renaissance, on p. 248 it talks about how they decided to set it at the time of Zefram Cochrane’s first warp flight:
It was then that lightning struck and they realized that by going back into The Next Generation’s past, they didn’t necessarily have to go back into the present day’s past—they could go into our future. And what event in our future is, perhaps, the most important event of Star Trek’s past?
There are two answers to that question: Cochrane’s first warp flight, and humanity’s first contact with aliens.
In thirty years of Star Trek story-telling, neither one of these pivotal events had ever been described in a filmed episode or motion picture—though it had been established in a fourth-season The Next Generation episode, also called “First Contact,” that another world’s first warp flight was the triggering mechanism by which the Federation made contact with a new race and revealed the existence of an interstellar civilization.
In other words, according to Star Trek’s own rules, Cochrane’s first warp flight and humanity’s first contact with aliens might be two events that were inextricably linked.
Moore is still surprised by how perfectly all these disparate elements suddenly meshed. "This was a section of Star Trek history that was left vague—purposefully vague, I'm sure—for many years. Gene had always said way back in the Original Series that there was a third world war at some point. And that somehow out of that came warp drive, came first contact with aliens, and came the Federation.
”It was embroidered a little bit over the years, but we were surprised when we went back and were looking at Star Trek history to see that we hadn't really loaded ourselves with a lot of detailed backstory.
“In 'Metamorphosis,' Cochrane's age was fortunately vague enough, and we looked through Mike and Denise Okuda's Star Trek Chronology, and all the dates came together very nicely. Warp drive and first contact could have taken place simultaneously, which made everything possible."
So, it seems that before they wrote this story, no date had been given for first contact, but as discussed on this page from the fan site Ex Astris Scientia, a speculative date of 2061 had been given for humanity's first warp flight in the first (1993) edition of the book Star Trek Chronology by Michael and Denise Okuda (Mike Okuda was the technical consultant for TNG along with Rick Sternbach, responsible for helping the writers maintain continuity, and his wife Denise collaborated with him on some of the reference books he wrote, and also worked on later series and movies after TNG). So, inspired by the TNG episode “First Contact” where it’s revealed the Federation has a policy of initiating first contact with civilizations when they first achieve warp flight, they decided that humanity too would have had their official first contact when Zefram Cochrane achieved his first successful test run of warp drive (the 1993 Star Trek Chronology also didn't say that Cochrane was actually on board the first warp ship, just that he was involved in building it, so this was another new element added by Braga and Moore). And since the Star Trek Chronology’s date of 2061 for the first warp flight was speculative, they had the freedom to bump it around a little for the movie (maybe they didn’t want to tread on the toes of Arthur C. Clarke’s 2061: Odyssey Three).
Why did the Okudas pick 2061 for the speculative date of Cochrane’s first warp flight? They explain in a note at the end of the 2061 entry in the 1993 edition of the Chronology that it was based on an episode of The Original Series (TOS) called "Metamorphosis", along with the estimated date of a ship called the S.S. Valiant that was featured in the TOS episode "Where No Man Has Gone Before". You can see the entry in the two google books snippets here and here, it reads:
First successful demonstration of light-speed propulsion by Zefram Cochrane. This became the basis for early warp drive technology.
Date is conjecture from Cochrane's career as described in "Metamorphosis," further assuming that early warp drive had to be invented before the S.S. Valiant embarked on its expedition, circa 2065, since that ship clearly crossed significant interstellar distances.
To see why these two episodes constrain the date to be somewhere in the vicinity of 2061 give or take a few years, first you need to know a little about the dating of TOS episodes. None of the episodes of TOS state what year the show was taking place, but the Okudas wanted a unified dating system for TOS and TNG, so it was decided (possibly by Roddenberry, as discussed in this answer) to have each TOS season be set 300 years after it first aired. This was later approximately confirmed on the show in the third season TNG episode “Sarek” where it was said that Sarek was 202 years old, and the TNG first season episode “The Neutral Zone” gave the date as 2364 so “Sarek” would be around 2366, and the second season TOS episode “Journey to Babel” had given Sarek’s age as 102.
So, the first constraint on the date came from the TOS episode “Where No Man Has Gone Before” which would be set around 2266 according to this assumption, though the date is given as 2265 according to Memory Alpha which presumably got this from the Chronology (maybe they set it in 2265 because it was a pilot episode and was filmed in 1965, plus it had a number of differences with other first season episodes, like different looking uniforms and a different chief medical officer). In that episode (transcript here) they discover the “disaster recorder” of the ship “S.S. Valiant”, which they say was ejected from the ship “two hundred years ago” after a magnetic space storm swept it "about a half light year out of the galaxy", and they say that it was able to make it back into the galaxy before being destroyed. Based on the name and the distances involved we can infer it was a warp-capable ship from Earth, and if “two hundred years ago” is taken as an exact figure (the Okudas said in Chronology that it was their general policy to assume numbers were exact unless this led to inconsistency), that would imply humanity already had warp drive in 2065 (this date for the Valiant was later confirmed in the Enterprise episode "In a Mirror, Darkly, Part II").
The second constraint on the date came from the TOS episode “Metamorphosis”, which had first introduced the character of Zefram Cochrane, telling us that he was the “discoverer of the space warp” and that he was thought to have died “a hundred and fifty years ago”, when his ship was lost while he was making a trip at age 87 (in the episode, he had been rejuvenated by an energy being called “the Companion”). This was a second-season episode from 1967, so according to the Okuda’s system it would be set in 2267. If you subtract 150 years that gives 2117 for when Cochrane disappeared, and if you subtract another 87 that would suggest he was born in 2030, and so would have been only 35 when the S.S. Valiant was lost. So probably the Okudas didn’t want to make him too much younger than this when he created warp drive (most geniuses aren’t Wesley Crusher style child prodigies after all), and that would have been the reason for setting the creation of the first warp ship in 2061 just a little before the date of the S.S. Valiant, when Cochrane would have been 31 according to these assumptions.
Incidentally, it may seem a little strange that Moore and Braga stuck with a date very close to this after the decision to hire the 56-year-old James Cromwell to play Zefram Cochrane for the 1996 movie, but maybe they figured the “one hundred and fifty years” comment from “Metamorphosis” could have been approximate rather than exact, so perhaps they imagined Cochrane could be a little older in their revised date of 2063 (as suggested by Moore's comment quoted earlier that "In 'Metamorphosis,' Cochrane's age was fortunately vague enough"). But later the Enterprise premiere episode “Broken Bow” showed a recording of Cochrane said to be from “from the dedication ceremony for the Warp Five Complex thirty two years ago” and Archer gave a specific date of “April 16th, 2151” in that episode, indicating the dedication ceremony took place in 2119, give or take a year. So even if he disappeared very shortly after that at age 87, his birth date would be in 2032, confirming that he was about 31 years old at the time of First Contact. Maybe the stress of living through the apocalypse caused premature aging?