Found on Wikipedia: Geronimo!

No, not Geronimo the Apache, not Geronimo Stilton, the mouse reporter, not Cesar Geronimo, outfielder for the Big Red Machine in the 70's, and not Sarah Geronimo, the cute young Filipina actress. Geronimo as in, "Geronimo!" Ever wonder why people say that when jumping off things?

According to Wikipedia, "Geronimo!" is believed to date from 1940, when an Army private was testing out a new invention called the parachute. Private Aubrey Eberhardt was scheduled to take a "mass jump" with the rest of his platoon, and, as you might expect, they were a little nervous about it.

To calm their nerves, the Army boys took in a movie--a western, in fact, which people now figure had to have been the 1939 film Geronimo with Andy Devine and Lone Ranger star Chief Thundercloud in the title role.

On the way back to barracks, Eberhardt said he expected the jump would be no different from usual. The others taunted him, saying that he would be too scared to remember his name. Eberhardt retorted, "All right, dammit! I tell you jokers what I'm gonna do! To prove to you that I'm not scared out of my wits when I jump, I'm gonna yell Geronimo loud as hell when I go out that door tomorrow!" Eberhardt kept his promise and the cry was gradually adopted by the other members of his platoon.

The expression stuck, and by 1941 the 501st Parachute Infantry Battalion had added the word "Geronimo" to its insignia.

By then the coverage of the paratroopers exploits during the war had made the cry "Geronimo" known to the wider public, and its use spread outside the military and air force.