Was the scene in which Marty travels from Twin Pines Mall in 1985 to the Peabody Farm in 1955 always meant to take place at 1.15 in the morning (1985 time), or was this a consequence of Eric Stoltz being fired and Michael J Fox being limited to mainly night-time shooting due to his commitment to Family Ties?
In the 4th draft of the screenplay, dated Oct 1984, the experiment takes place at 1AM:
MARTY: What's up, Doc?
BROWN: One a.m., right, Marty? You're gonna be there, right? Twin Pines Mall.
MARTY: Yeah, right.
This is definitely from before shooting started -- at this point, "Suzy" had not yet been renamed "Jennifer". Therefore, it was definitely before Eric Stoltz was replaced.
I've not found any indication why the script says 1:15 but we can infer a few reasons in-universe
- Doc Brown was a very precise man (or a geek in modern vernacular). Remember the exactly 25 minutes slow clocks in the opening of the movie? The specific time was in keeping with the character being notably eccentric.
- The mall parking lot was virtually guaranteed to be empty (malls were social gathering places in 1985) to allow them the freedom to hit 88MPH and not attract any attention
- Brown was trying to evade the Libyans
Out of universe, Michael J Fox was only free to work nights
The only reason Michael J. Fox was even able to appear in Back to the Future was that his Family Ties co-star Meredith Baxter was getting ready to give birth to twins. The show’s schedule had to be changed as a result, and Fox was allowed to work part-time on Back to the Future.
During the nine weeks of shooting, the film crew mostly got to work with Fox only at night, as he would spend most of his days in Paramount rehearsing that week’s episode of Family Ties. In the evening, a driver would take him to whatever location Back to the Future was filming at that night. Fox would work on the movie until early in the morning, when he would be taken back home to catch a couple of hours of sleep. Then, another driver would wake him up, make him a coffee and take him to Paramount for another day of rehearsing.
In universe, the novelization (according to the BttF wiki) attributes it to Doctor Brown not being comfortable around any kinds of crowds, and wanting to avoid any people that might be at the mall for other reasons (Such as the movie theater letting out).
"Doc Brown eased the venerable step-van onto the Twin Pines Mall parking lot shortly after midnight. There were more cars than he expected, so he pulled to the far rim of the asphalt area and waited. "Must be a long movie," he said to himself. "
—From Back to the Future by George Gipe (quote, page 41)
"A few minutes later, several dozen people emerged at once from the mall's interior and moved to their cars. A series of starting engines, blinking lights and squealing rubber enlivened the vast treeless plain for a few minutes. Then all was silence again. The faint smell of gasoline fumes hung in the thick air as the tiny specks of light disappeared into the morning darkness. In comparative solitude once again, Brown felt better. People made him feel vaguely insecure."
—From Back to the Future by George Gipe (quote, pages 41 and 42)
Admittedly, that explains why late at night, but not 1:15 am specifically.