To grab a random object from an array in javascript you can use:
var randomItem = arr[Math.floor(Math.random()*arr.length)]
Combining this with your request you can do:
;(function($, window, document, undefined){
$(function(){
var images = ['01.png', '02.png', '03.png'];
$("[rel='apple-touch-startup-image']").attr("href", images[Math.floor(Math.random()*images.length)]);
});
})(jQuery, this, document);