Buddy Holly said that line from “The Searchers” inspired him to write that song.
From wiki:
"In June 1956, Holly along with his older brother Larry as well as Allison and Sonny Curtis went to see the film The Searchers, starring John Wayne, in which Wayne repeatedly used the phrase "that'll be the day". This line of dialogue inspired the young musicians."
Here’s a better explanation from another News Source:
“In 1956 Buddy Holly and his drummer Jerry Allison went to see a movie in their hometown of Lubbock, Texas. They saw "The Searchers" starring John Wayne. Wayne had a response that he used throughout the movie when someone said something he didn't agree with: "That'll be the day."
“As legend has it, after the movie, Holly looked at Allison and said that it sure would be nice if they could record a hit song. And Jerry replied "That'll be the day."
That was the inspiration for a song that was No. 1 on the Billboard pop chart 60 years ago.”
