Adding a Search in the header?

Why don’t put a widget area above social+tagline+menu? You might want to add other widgets there..
If you like this idea that’s what you have to do:
In your child theme functions.php:

/*Create a new widget area*/
if (function_exists('register_sidebar')) {
    register_sidebar(array(
    'name' => 'Navbar Top Widget Area',
    'id' => 'extra-navbar-top-widget-area',
    'description' => 'A widget area above social block and tagline in header',
    'before_widget' => '<div id="%1$s" class="widget %2$s">',
    'after_widget' => '</div>',
    'before_title' => '<h2 class="widget-title">',
    'after_title' => '</h2>'
    ));
}
/* Add the new widget area into navbar-inner before social-block*/
function add_navbar_top_widget_area(){
    if ( function_exists('dynamic_sidebar') ){
        ob_start();
        ?>
        <div id="navbar-top" class="widget-area span12">
            <?php dynamic_sidebar('Navbar Top Widget Area'); ?>
        </div>
        <?php
        $html = ob_get_contents();
        ob_end_clean();
        echo $html;
    }
}

Now you can add the search widget to this area and of course you have to style it in your child theme style.css .
Note that now social-block, is not the first child of the row anymore, so you have to set its left margin to 0px, this way:

.navbar-inner .social-block {
    margin-left: 0px;
}

Hope this helps

Ops, I missed an important part, sorry, copy&paste error.
Before this: function add_navbar_top_widget_area(){
add this:

add_action('__navbar', 'add_navbar_top_widget_area', 1);

You can start with this to hide the form and display it on hover:

/* don't display the label "Search" */
#navbar-top .screen-reader-text{
    display: none;
}
/* Hide the form */
#navbar-top .searchform{
    display: none;
}
/* Use the magnifying glass as icon*/
#navbar-top .widget_search:before{
    content: "\f400";
    font-family: "genericons";
}
/* When hover over the div widget_search display the form*/
#navbar-top .widget_search:hover .searchform{
    display:block;
}

And of ‘course you can style your form and the widget the way you want just like you did with everything else.

why do you say so? Looks like it works.

Glad to hear that.

chappie, don’t know if I really understand :D, sorry.
can you give me a link to your site?

Sorry don’t have a mac.
So let me understand the issue, you use my snippet above to place the search above the navbar. You want it aligned to the right top of the slider, which I presume, is in non full width mode (eheh easy assumption 😀 ). You have a disappearing problem when resizing the window, right?
Let me try if I can reproduce it or not.

You’ve got it 🙂 Thanks for taking a look.