An entry from the Dictionary of Word and Phrase Origins, Second Edition by William and Mary Morris:
From the earliest wars in recorded history, men have plunged into battle shouting battle cries. Indeed, our common word 'slogan' was originally the Gaelic sluggh-ghairm, meaning the call to battle used by Scottish Highlanders and Irish clan. One of the most interesting of these cries is that used by the U.S. airborne paratroopers: "Geronimo!"
When we speculated in print on why our soldiers use the name of a dead Apache chieftain for their slogan, several alumni of airborne regiments reported stories of its origin. A plausible one came from Arthur A. Manion. "At Fort Sill, Oklahoma," he wrote, "a series of rather steep hills, called, I believe, Medicine Bluffs, was pointed out to all new arrivals. It was said that one day Geronimo, with the army in hot pursuit, made a leap on horseback down an almost vertical cliff Ã‚ a feat that the posse could not duplicate. The legend continues that in the midst of this jump to freedom he gave out the bloodcurdling cry of "Geronimo-o-o!" Hence the practice adopted by our paratroopers. I hope this helps. It's at least colorful, if not authentic."
Another correspondent, who once lived at Fort Sill, added the information that the bluff from which Geronimo made his daring leap "is a cliff overlooking a small river." So we know that Geronimo and his steed had water, rather than desert floor, to break their fall. Now, this is indeed an interesting tale and one that may very well be the real inspiration for the paratroopers were trained at Forts Bragg and Campbell. Why, then, did they reach to Fort Sill for inspiration for their battle cry?
R. Collier of Milwaukee offered a less glamorous but probably more accurate account of the origin of the call. "In the early days of the 82nd Airborne," he wrote, "the men used to go to the nearby movie in Lafayetteville. During the week scheduled for the division's initial jumps, they saw a movie named Geronimo. (If that wasnÃ‚'t the title, at least the Indian chief played a leading part.) Anyway, one guy hollered the name and one of those things no one can explain happened. The whole division took it up and from them it spread to the later-activated airborne forces."
An entry from the Glossary Britannica.com :
Indian name Goyathlay ("One Who Yawns")
Bedonkohe Apache leader of the Chiricahua Apache, who led his people's defense of their homeland against the military might of the United States.