It was Mayor George B. McClennan (son of the Civil War general) who announced the official name change. "Times" was, of course, drawn from the New York Times, the newspaper which had recently moved into a building perched on a traffic triangle in the middle of the square. The daily, which lobbied hard for the name change, would soon move to larger quarters on 43rd Street, where it still operates today.
The Times owner and publisher, Adolph Ochs, knew what he was doing. The first segment of the city's subway system—of which the Times Square stop would be the major transfer location—opened later that year, and soon transformed the neighborhood into a bustling center of business and entertainment.
The Times also, in 1904, inaugurated the tradition of an open-air welcoming of New Year's Eve.
The thing the area eventually became best known for, of course, was theatre. The stage got a jump on the New York Times, as far laying a claim to the neighborhood's booming future. The Casino, the very first theatre in what would become the city's latest and most lasting theatre district, opened in 1882. It was followed by the Broadway, Empire, American, Abbey's, Olympia, Victoria, Republic, Circle, Majestic, Lyric, Lyceum, Hudson, and New Amsterdam—all of which opened before the name Times Square was coined.
The Republic, Lyceum, Hudson and New Amsterdam still stand. (Parts of the Lyric were incorporated into the Ford Center for the Performing Arts.) The Republic was reborn as the New Victory, and the New Amsterdam was reclaimed by Disney, both after suffering decades of neglect and misuse. The Hudson hasn't been used as a legitimate stage since 1968. Among Broadway theatres who knew both Longacre and Times Squares, only the Lyceum has seen constant use as a theatre. The nostalgically named Longacre Theatre was built in 1913.