"Cocky" is the name of the mascot for the University of South Carolina Gamecocks since 1980. The previous mascot was the character "Big Spur," supposedly Cocky's father. Cocky is a red gamecock/rooster with black legs and a black jersey (garnet and black are the university's colors) and a yellow beak and legs. Apparently Cocky won some sort of mascot competition twice since he's become the USC mascot: the university's athletic website says he "captured National Championship titles as the No. 1 mascot in 1986 and 1994." In February 2002, he even appeared in an Anheuser-Busch ad with other mascots, urging people not to drink and drive. "Old World Christmas" even offers a figurine of Cocky making a snowman who resembles himself.

So, doesn't USC realize how innuendo-laden references to Cocky and the Gamecocks can become? Probably. Though the Gamecocks name for the teams dates back to the early 1900s, "Cocky" was named recently enough that the modern connotation of "cock" was probably considered. I have seen bumper stickers on cars in Columbia, South Carolina that proclaimed "God must love Carolina 'cause he made so many Cocks!" A current South Carolina resident's website records such merchandise as:

"even the simple black shirt with white lettering, reading, "You Can't Lick Our Cocks." Hats, notebooks, shirts, pins, bumper stickers, and so many more things are covered with these phallic sayings. Even the scuba club and football team have gotten into this with their own shirts saying such things as "Wet Cocks Go Deeper--Always Go Down With A Buddy" and "Iron Cocks." As one of my friends pointed out, if USC (my USC) went up against USC (University of Southern California) it would be a VERY phallic game--the Cocks versus the Trojans. Personally, I would love to see that game, if for nothing more than for the banners each side could come up with to cheer their team on. Maybe it's something that both universities should look into for a good athletics fundraiser."
So apparently fans find the double entendre amusing. (PETA's brief campaign in 2001 to persuade the school to call its teams something besides the Gamecocks probably made most fans even stubborner about keeping the old name. PETA's letter was placed with other 2001 memorabilia in a time capsule to be opened in 2051.)

Sources:
http://uscsports.ocsn.com/ot/spirit/scar-spirit-cockys-corner.html
http://usccocks.rivals.com/content.asp?CID=85421
http://www.jca4usc.com/traditions.htm
http://rseanes.com/wrong/ACL-usc.html
http://www.old-world-christmas.com/soutcarmassn.html
http://espn.go.com/magazine/geno_20011028.html
http://www.sc.edu/bicentennial/pages/timecapsulecontents.html