As probably one of the longest lifers around, I do not think you can pick one "greatest" athlete. There have just been too many, playing in different eras and in different sports. It is really difficult to compare, say, Bobby Layne with Vince Young. Both would qualify as great athletes at Texas, but they are from different times. Kids today are just bigger, stronger and faster than they were during Layne's time. Back then there was no weight training (that was actually discouraged). Concepts and style of play has also drastically changed. Layne was Texas' first "T" quarterback.....a position that was just being implemented in football. Before he played his senior year, Texas was a single wing team, and the "T" was adopted to take advantage of his athleticism. Of course, Bobby was a two sport guy....he was an outstanding pitcher.
By the same token, it is really difficult to compare Tom Hamilton and Slater Martin with guys like Kevin Durant and T.J. Ford. Different eras, different game.
Don't forget the guys who were setting world records in the sprint relays back in the 1950s. Toward the end of that run we had a kid named Eddie Southern, who was extraordinary.
There have been great baseball players and golfers as well.
I won't take the time, but there are a hundred or more guys that could go on a list of the greatest....all of them great players in their time. Trying to pick one will just boil down to a popularity contest, because there are so many that would qualify....some relatively unknown to most.