To change your site URL, go to your Account Settings > Me tab. In the Account section, look for the Site URL field at the top. Type in whatever you'd like your new site URL to be, then click Save in the header toolbar to ensure your changes are saved.
Your pictures and everything else on your site will remain the same, but your website URL will be different. For example, if your site URL was previously "myphotos," your old URL was myphotos.smugmug.com and will no longer work. Your new URL will be mynewnickname.smugmug.com.
Don't forget to let your loyal fans know your website address has changed!