Pulling a variable into the wp_nav_menu function

I think your scope is not the same. You are defining the $menu_name in one scope and calling it in another scope. So you can do it with global variable like below-

$GLOBALS[ 'menu_name' ] = 'Section 1 Menu';

And call it like-

<div id="nav">
    <div class="close-menu"></div>
<?php
wp_nav_menu(
    array(
        'menu'              => $GLOBALS[ 'menu_name' ],
        'theme_location'    => 'primary',
        'depth'             => 1,
        'menu_class'        => 'main-menu',
        'menu_id'           => 'main-menu'
    )
); ?>
    <div class="header-tools">
        <div class="icon-list">
            <a href="https://wordpress.stackexchange.com/help" class="icon help trigger-help" title="help"></a>
            <a href="http://wordpress.stackexchange.com/dashboard" class="icon account" title="my account"></a>
        </div>
    </div>
</div>

Hope that helps.