Javascript event handling via switch
This sort of reminds me of how React implements Flux, where there's a huge switch list, and it becomes very unwieldy fast. Another developer created the Reflux library using the same concept in your alternative, a huge action map. So you're getting there. Let's just tweak your approach a bit.

First is you can use the noop approach, so you avoid the if and the function is always a function. It's pretty functional programming, but it streamlines your code, avoiding branching, unnecessary assignments and all that stuff.

Next, functionObj could be named better, like registeredFunctions. Also create an interface for you to add easily, as well as check for duplicates to prevent collision, or just register multiple ones under the same name.

Next is to allow your callers to define multiple args by not limiting the args. Here, we pass in arguments using apply. That way, the callers can provide any number of args they want and the receiving functions can still receive them. Of course, we slice off the first argument, which is your string of functions. You can assign any value as context for your registered function, but I prefer functions like these to have a context of null. Context becomes confusing when calling functions dynamically, and without a set convention.

var noop = function(){}
var registeredFunctions = {/* map of functions */}

function process(){
  var args = Array.prototype.slice.call(arguments, 1);
  functionString.split(";").forEach(function(functionName){
    (registeredFunctions[functionName] || noop).apply(null, args);
  });
}


An alternative would be to keep the huge switch, but instead of putting the actual code inside it, you call a function. There, you have a new scope. You just need to pass in everything you need into the function.

var args = Array.prototype.slice.call(arguments, 1);
switch(something){
  case 1:
  case 2:
    doSomething.apply(null, args);
    break;
  case 3:
    doSomethingElse.apply(null, args);
  default:
    doSomethingDefault.apply(null, args);
}