Here's one way

myString.split('').map(function(c){
    return c[Math.round(Math.random())?'toUpperCase':'toLowerCase']();
}).join('');

      

        
        
        
      

    

You can add this as a method prototype for a string object for ease of access:

String.prototype.toRandomCase = function() {
    return this.split('').map(function(c){
        return c[Math.round(Math.random())?'toUpperCase':'toLowerCase']();
    }).join('');
}

      

        
        
        
      

    

Then access to

console.log(myString.toRandomCase());

      

        
        
        
      

    

A little about how it works:

• String.split
  
  used to split a string into an array of individual characters. Function used
• Array.map
  
  ... This executes a callback function that is applied to each item in the array and returns a new resulting array with each value returned by the display function.
• Inside the map function
  • Math.round(Math.random())
    
    used for randomness
  • The result of this is used with the ternary operator to get toLowerCase
    
    ortoUpperCase

Math.Round(Math.random())?'toLowerCase':'toUpperCase'
    
    
  • The result of the ternary operator is used to access the corresponding property of the function by backtracking the character array and then calling. c[<ternary here>]()
    
    
• Finally, it uses a method Array.join
  
  on the result of a function call map
  
  to concatenate the resulting array back into a string.

Code golf (efficiency)

RobG's answer has a more efficient approach than mine (please support his answer)

String.prototype.toRandomCase = function() {
    return this.toLowerCase().split('').map(function(c){
        return Math.random() < .5? c : c.toUpperCase();
    }).join('');
}

      

        
        
        
      

    

If anyone has any suggestions for improving this, please comment or edit this part of the answer