Try setting an expiry date on the cookie. Code below sets it one year in the future.
a = new Date(new Date().getTime() +1000*60*60*24*365);
document.cookie = 'mycookie=somevalue; expires='+a.toGMTString()+';';
Please give it a shot and check the resources
tab again to see if it's changed.
Here is some info regarding cookie syntax and options.