Seems that onBlur of one element is "overriding" the onclick of another

I think you are short on answers because your question is confusing. Presumably you have an input that, when focussed, shows a list of suggestions based on the characters entered into the input.

If the user uses the cursor to select an item, then I suppose the blur event of the input fires before the click event of the div and the the div is set to display:none before the click fires, and hence misses the div.

The fix is to call the onblur listener after a short timeout, so:

  <input ... onblur="setTimeout(function(){hideSelect();}, 100);">

Test in a number of browsers, you may need to set the timeout to 200ms or so. It doesn't matter if there's a short, visible delay after the blur event before the suggestions disappear (i.e. a bit too long is better than a bit too short).

Make sure the suggestions don't obscure anything important on the page or users may find them more of a hindrance than a help. :-)

The accepted answer will work as a quick fix, but by relying on a setTimeout, you assume that the user will only keep clicking down for less than n milliseconds before they release (just picture someone hesitating on a click). To be more sure that the click will go through, you could set a longer timeout, but that means your hide-able element will stay visible that much longer after the blur.

So let's look at the root of the problem.

The click event failing to go through is a result of the events being fired in the following order:

1. mousedown
2. blur
3. mouseup
4. click

So by the time the mouseup/click events are ready to fire, the blur listener has been called and the element you had once been hovering over has already disappeared.

Here's a general fix (based on the fact that the mousedown event fires first) that should work:

var searchEl = $('#search');
var listEl = $('#dropdown');
var keepListOpen = false;

searchEl
  .on('focus', function() {
    listEl.show();
  })
  .on('blur', function() {
    // Hide the list if the blur was triggered by anything other than
    //  one of the list items
    if (!keepListOpen) {
      listEl.hide();
    }
  });

listEl.find('li')
  .on('mousedown', function(event) {
    // Keep the list open so the onClick handler can fire
    keepListOpen = true;
  })
  .on('click', function(event) {
    // Proof that the list item was clicked
    alert('clicked option');
  });

$(window).on('mouseup', function(event) {
  // Return the keepListOpen setting to its default and hide the list

  // *NOTE* We could have tied this handler to the list items, 
  // but it wouldn't have fired if a mousedown happened on a
  // list item and then the user dragged the mouse pointer 
  // out of the area (or out of the window)
  if (keepListOpen) {
    listEl.hide();
    keepListOpen = false;
  }
});

// Bind to `window.top` if your page might be displayed in an iframe
// $(window.top).on('mouseup', function(event) {
//  if (keepListOpen) {
//    listEl.hide();
//    keepListOpen = false;
//  }
//});

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<input id="search" type="text" autocomplete="off" placeholder="Click Here">

<ul id="dropdown" style="display: none;">
  <li>Click Me 1</li>
  <li>Click Me 2</li>
  <li>Click Me 3</li>
</ul>