Change Parent Name with wp_list_pages?

You could of course use some preg_replace() tricks to solve this, but here’s a little (untested) idea using the the_title filter instead:

add_filter( 'the_title', 'wpse_title' ); 
$children = wp_list_pages("title_li=&include=".$post->post_parent."&echo=0");    

where our filter callback is:

function wpse_title( $title )
{
    remove_filter( current_filter(), __FUNCTION__ );
    return __( 'Overview' );
}

Remark 1:

I think you should consider replacing the wp_list_pages() method of generating the single parent link and instead use get_permalink( $parent_id ) or get_page_link( $parent_id ) from the parent page ID. Then you can easily from control the link name.

 <li>
      <a href="https://wordpress.stackexchange.com/questions/156742/<?php echo get_permalink( $parent_id ); ?>">Overview</a>
 </li>

You already use this for one of the levels, so why not for others as well?

Remark 2:

You could also use get_post_ancestors() to help you get the current tree level position.

For example:

$ancestors_ids = get_post_ancestors( get_the_ID() );
$level         = count( $ancestors_ids ) ;

Then you can get the parent ID, of the current page, with:

$parent_id = ( $level > 0 ) ? array_shift( $ancestors_ids ) : 0;

Here’s an example of how the case $level = 2 could look like:

<ul class="submenu">
    <li>
        <a href="https://wordpress.stackexchange.com/questions/156742/<?php echo get_permalink( $parent_id ); ?>">Overview</a>
    </li>
    <?php echo wp_list_pages("title_li=&child_of=".$parent_id."&echo=0&depth=1"); ?>
</ul>

Maybe your $level = 1 could then be:

<ul class="submenu">
    <li>
        <a href="https://wordpress.stackexchange.com/questions/156742/<?php echo get_permalink( $parent_id ); ?>">Overview</a>
    </li>
    <?php echo wp_list_pages("title_li=&child_of=".get_the_ID()."&echo=0&depth=1"); ?>
</ul>

etc ..