Why are the Giants called the Giants?

From the Giants own history section of their web site, "On June 3, after a rousing extra-innings victory over Philadelphia, manager Jim Mutrie was so overcome with emotion that he supposedly blurted out a description of his team that immediately became the franchise's new nickname. He called them his Giants."

There have also been rumors that they were called the Giants because many of the players were tall and because there were many skyscrapers in New York, but the above is the truest story.