"Go ahead, make my day" was a line said by Clint Eastwood as the movie character "Dirty Harry" Callahan, a San Francisco police inspector, in the Dirty Harry series of movies that he made (specifically, Sudden Impact).
On the Sunday morning before his gunfight with Buford Tannen, Marty McFly practiced being tough in front of a mirror, and did his best Dirty Harry imitation of the real Clint Eastwood. He then walked into town, where everyone greeted him as "Mr. Eastwood".
On June 13, 1931, Edna Strickland during her interview with Marty asked if he wanted to give a message to the criminal element of Hill Valley. Marty responded "Go ahead, make my day". Edna asked "Make your day what?"