Why did parachutists yell “Geronimo”? Or is that just a Hollywood invention, but–even so–why “Geronimo”?
These victories spurred the creation of the U.S. airborne program, and a fifty-man test platoon was formed on June 25, 1940. On August 16, Lieutenant William Ryder became the first member of the test platoon to make a parachute jump. In one demonstration jump, the famous airborne phrase “Geronimo!” was born when Private Aubrey Eberhardt yelled it after exiting the plane.
From here.
And here.
See also Uncle Cecil’s answer from Triumph of the Straight Dope: Why do parachutists yell “Geronimo!” when jumping from an airplane?
From what I’ve heard, at one point in the movie Geronimo leaps down upon an enemy shouting his own name. Private Eberhardt apparently copied this. Not having seen the film myself, I can’t swear that any such scene exists, but that’s what I heard.
Well, not really, otherwise he would have shouted “Eberhardt!” 
I think it was in one of the Charlie Sheen “Hot Shots!” movies that several paratroopers jump out of a plane, shouting the requisite name, followed by a man in full Indian regalia who shouts, “Me!”