what is the significance of the elephant on the sleeve patch?

Answer 1

This is taken from a site that describes the reason better than I could:

"In 1901 Connie Mack and his Philadelphia Athletics became one of the original founders of the American League. In 1902 New York Giants Manager John McGraw dismissed the A's with contempt, by calling them "The White Elephants." He meant to imply that Mack shouldn't be allowed to spend money without supervision. Well, Connie Mack took up the gauntlet and defiantly adopted the White Elephant as the team insignia. That year, 1902, the A's won the American League pennant -- much to the unvoiced chagrin of John McGraw."

Answer 2

I assume you are talking about the Oakland A's.... This is from wikipedia:

"After New York Giants' manager John McGraw told reporters that Philadelphia manufacturer Benjamin Shibe, who owned the controlling interest in the new team, had a “white elephant on his hands," Mack defiantly adopted the white elephant as the team mascot, though over the years the elephant has appeared in several different colors (currently forest green). The A’s are sometimes, though infrequently, referred to as the Elephants or White Elephants."

"The elephant was retired as team mascot in 1963 by then-owner Charles O. Finley in favor of a Missouri mule. In 1988, the elephant was restored as the symbol of the Athletics and currently adorns the left sleeve of home and road uniforms. In 1997, the elephant came to life for the first time in the form of a mascot, Stomper, whose likeness was patterned somewhat after A's slugger Mark McGwire."