The generally accepted theory is that America was named after the Italian explorer, Amerigo Vespucci...just as Boo's link above says.
That's one story...here's another. The British writer, Rodney Broome, recently published a book called Amerike, the Briton Who Gave America its Name. It is based on research he carried out in the archives of Westminster Abbey. The key finding he made there was that both Vespucci and Columbus had used maps created by John Cabot.
He had made voyages to the New World from Bristol, England, and made detailed maps. These trips were sponsored by King Henry VII and a Welsh aristocrat called Richard Amerike. (Merrick or Maereg is still a common Welsh Christian name.) He christened the new land �America' in honour of his patron in 1496...Amerike/America...before Vespucci had ever even seen it!
Because the fame of Vespucci and Columbus grew and grew while Cabot's faded, the mistaken belief arose that they had named the new land.
So, there are two theories. Take your pick.