Improving JavaScript - using objects vs. just functions
If this is already your entire code, then you won't benefit much from namespacing your functions. Namespacing is basically assigning stuff to an object property. Accessing object properties is a minor overhead, but an overhead nevertheless. 

Also, it doesn't look like anything else is accessing the code made for click and hover, so I'd rather place them directly into the handler instead.

You should be fine with simple functions in this case.

I'd rather concentrate more on reducing redundant code. Here's what I have done:

//$(document).ready(fn) can be shortened to $(fn)
$(function () {

  //declare variables in a function block up top
  //this is how the compiler sees them, so to avoid unexpected behavior
  //just follow suit
  var open = false
    , peek = false
    , helpBar = $(".help-bar")
    , helpIcon = $(".help-icon")
    , help = helpBar.add(helpIcon)
    ;

  //as explained, namespacing isn't an advantage at this level of code
  //and so, I have removed the functions and placed the code into the handlers

  //you can modify "this" in a function by using .call() or .apply()
  //in this case, element argument is removed and will be passed as "this"
  //this method will be explained later
  function changeRightValue(amount, speed) {
    this.animate({
      right: amount
    }, speed);
  }

  function toggleArrow(direction) {
    helpIcon.css("background", "url('/images/buttons/zapp/arrow" + direction + ".png') no-repeat 0px 50%");
  }

  //here we have our merged functions

  helpBar.on('click',function () {

    //if blocks that call the same functions but differ in parameters are messy
    //what I did here is made ternary ifs to pick arguments, not pick invocations
    //the calls will be made later
    var changeRightParams = open ? [0, 'slow'] : [280, 'fast']
      , direction = open ? 'Open' : 'Close'
      //this is a very short way to toggle booleans in the same variable
      , open = !open
      ;

    //here we call the functions with the "picked" arguments

    //here we use .apply to set the invoked function's this as help
    //and the arguments as the picked arguments
    //it's the same as calling "help.changeRightValue(amount,speed)
    changeRightValue.apply(help, changeRightParams);
    toggleArrow(direction);

  }).on('hover',function () {

    //like above, declare variables up top to avoid unexpected behavior
    var changeRightParams;

    //here we use the "early return" pattern, to avoid nesting ifs
    //since nothing executes when open is true, we return early
    //we saved 1 indent level in the code
    if (open) return;

    //code after the early return assumes "open" is false
    //it would be similar to if(!open){/*stuff here*/}

    help.stop();

    //same as above, we pick arguments, toggle the boolean, call with the selected
    changeRightParams = peek ? [0, 'fast'] : [5, 'slow'];
    peek = !peek
    changeRightValue.apply(help, changeRightParams);

  });
});