Answer by taking a quote from the following article:

Much of the information about the early years of Calamity Jane's life comes from the autobiographical booklet which she dictated in 1896, written for publicity purposes. She was about to begin a tour in which she appeared in dime museums around the United States, and it was intended to help attract audiences. Some of the information in the pamphlet is exaggerated or even completely inaccurate. Calamity Jane was born on May 1, 1852 as Martha Jane Canary (or Cannary) in Princeton, within Mercer County, Missouri.

Jane was involved in several campaigns in the long-running military conflicts with American Indians. Her claim was that:  It was during this campaign [in 1872-73] that I was christened Calamity Jane. It was on Goose Creek, Wyoming where the town of Sheridan is now located. Capt. Egan was in command of the Post. We were ordered out to quell an uprising of the Indians, and were out for several days, had numerous skirmishes during which six of the soldiers were killed and several severely wounded. When on returning to the Post we were ambushed about a mile and a half from our destination. When fired upon Capt. Egan was shot. I was riding in advance and on hearing the firing turned in my saddle and saw the Captain reeling in his saddle as though about to fall. I turned my horse and galloped back with all haste to his side and got there in time to catch him as he was falling. I lifted him onto my horse in front of me and succeeded in getting him safely to the Fort. Capt. Egan on recovering, laughingly said: "I name you Calamity Jane, the heroine of the plains." I have borne that name up to the present time.  Captain Jack Crawford served under Generals Wesley Merritt and George Crook. According to the Montana Anaconda Standard of April 19, 1904, he stated that Calamity Jane "never saw service in any capacity under either General Crook or General Miles. She never saw a lynching and never was in an Indian fight. She was simply a notorious character, dissolute and devilish, but possessed a generous streak which made her popular."  It may be that she exaggerated this story, or even completely fabricated it. Even during her lifetime, not everyone accepted her version as true. A popular belief is that she instead acquired it as a result of her warnings to men that to offend her was to "court calamity". It is possible that "Jane" was not part of her name until the nickname was coined for her. It is certain, however, that she was known by that nickname by 1876, because the arrival of the Hickok wagon train was reported in Deadwood's newspaper the Black Hills Pioneer on July 15, 1876, with the headline: "Calamity Jane has arrived!"  Another unverified story in her autobiographical pamphlet is that her detachment was ordered to the Big Horn River under General Crook in 1875. She swam the Platte River and travelled 90 miles (140 km) at top speed while wet and cold in order to deliver important dispatches. She became ill afterwards and spent a few weeks recuperating. She then rode to Fort Laramie in Wyoming and joined a wagon train headed north in July 1876. The second part of her story is verified. She was at Fort Laramie in July 1876, and she did join a wagon train that included Wild Bill Hickok. That was where she first met Hickok, contrary to her later claims, and that was how she happened to come to Deadwood.

Why was she given this nickname?
"I name you Calamity Jane, the heroine of the plains."