次のようなことを試してください:
jQuery(document).ready(function($){
var a=[];
a[0]= [["1","2","3"],["1","2","3"],["1","2","3"]];
a[1]= [["8","9","9"],["5","6","7"]];
var json_string_old=JSON.stringify(a);
//set myCookie
$.cookie('myCookie',json_string_old);
//change the value of the array as per your requirments
a[1]=[["2","2","2"],["2","2","2"]];
a[2]=[["4","4","4"],["4","4","4"]];
//convert it again to JSON String
var json_string_new=JSON.stringify(a);
//now compafre & update accordingly
if(json_string_new === json_string_old){
console.log("same no need to update cookie");
}else{
console.log("different ok lets update the cookie");
$.cookie('myCookie',json_string_new);
}
});