Every object has a method toJSON(), which is invoked when the object should be serialize using JSON.stringify().  
From MDN article on JSON.stringify():
If an object being stringified has a property named toJSON whose value
  is a function, then the toJSON() method customizes JSON
  stringification behavior: instead of the object being serialized, the
  value returned by the toJSON() method when called will be serialized
The following example creates a different serialization function, which will exclude the _id:
var Player_ID = 0;
function Player(name) {            
 this._name = name;                     
 this._id = "Player"+(Player_ID++);
 this.toJSON = function() {
   return {
     _name: this._name
   };
 };
};
var newPlayer = new Player('Name 1');
console.log(JSON.stringify(newPlayer)); // prints {"_name": 'Name 1'}
Check the working demo.