[Faab]

Попробуйте так:

function openbox(id){
display = document.getElementById(id).style.display;
  if(display=='none'){
    document.getElementById(id).style.display='block';
    localStorage["menuLeft"] = "block";
  }else{
    document.getElementById(id).style.display='none';
    localStorage["menuLeft"] = "none";
  }
}

window.onload = function(){
  if(localStorage["displayMenuLeft"] != "undefined"){
    document.getElementById(id).style.display = localStorage["menuLeft"];
  }else{
    localStorage["displayMenuLeft"] = "block";
  }  
}