[WordPress] Custom site titles for the My Sites network admin bar menu in a multisite network. Configure first function, then auto-activate by saving this file in wp-content/mu-plugins, or network activate as a regular plugin.

multisite-custom-my-sites-menu-titles.php

<?php
/**
 * Plugin Name:       Custom My Sites Menu Titles
 * Description:       Custom site titles for the My Sites network admin bar menu. Multisite only.
 * Version:           2015.11
 * Author:            Caspar Hübinger
 * Author URI:        https://profiles.wordpress.org/glueckpress
 * License:           GNU General Public License v3 or later
 * License URI:       http://www.gnu.org/licenses/gpl-3.0.html
 */


if ( ! defined( 'ABSPATH' ) ) {
	exit;
}

/**
 * THIS NEEDS TO BE ADJUSTED TO YOUR NEEDS!
 *
 * @return array  Site IDs and site title customizations
 */
function ms_custom_site_menu_titles__custom_settings() {
	/*
	 * USE SITE IDs FROM YOUR NETWORK HERE!
	 * Schema: existing site ID => new site title
	 */
	$your_network = array(
		1  => array( 'site_menu_title' => 'Site DE' ),
		70 => array( 'site_menu_title' => 'Site AT' ),
		42 => array( 'site_menu_title' => 'Site CH' ),
	);

	return $your_network;
}


/**
 * Initialize plugin.
 *
 * @return void
 */
function ms_custom_site_menu_titles__load() {

	// Logged in multisite users only.
	if ( ! is_user_logged_in() || ! is_multisite() )
		return;

	// Remove core version of My Sites menu.
	remove_action( 'admin_bar_menu', 'wp_admin_bar_my_sites_menu', 20 );

	// Add custom copy of My Sites menu, including new filter.
	add_action(
		'admin_bar_menu',
		'ms_custom_site_menu_titles__admin_bar_my_sites_menu',
		21
	);

	// Filter and modify site titles for My Sites menu.
	add_filter(
		'ms_custom_site_menu_titles__admin_bar_network_site_title',
		'ms_custom_site_menu_titles__custom_site_menu_titles',
		10,
		4
	);

}
add_action( 'plugins_loaded', 'ms_custom_site_menu_titles__load' );

/**
 * Assign custom site titles for network sites.
 *
 * @param  string $site_title  Default menu item text
 * @param  string $blavatar    An empty div by default
 * @param  string $blogname    Site title
 * @param  string $userblog_id Site ID
 * @return string              Modified or default menu item text
 */
function ms_custom_site_menu_titles__custom_site_menu_titles( $site_title, $blavatar, $blogname, $userblog_id ) {

	// Fetch custom network settings from top of file.
	$your_network = ms_custom_site_menu_titles__custom_settings();

	// Sanitize filter argument.
	$current_site = absint( $userblog_id );

	// Assign new site title.
	if ( isset( $your_network[ $current_site ] ) ) {
		$site_title = $blavatar . $your_network[ $current_site ][ 'site_menu_title' ];
	}

	return $site_title;

}

/**
 * Exact copy of the "My Sites/[Site Name]" menu as in WordPress 4.4-beta2,
 * except 1 filter has been added (ms_custom_site_menu_titles__admin_bar_network_site_title).
 *
 * @see    wp-includes/admin-bar.php
 * @param  object $wp_admin_bar Admin bar object
 * @return void
 */
function ms_custom_site_menu_titles__admin_bar_my_sites_menu( $wp_admin_bar ) {

	// Don't show for logged out users or single site mode.
	if ( ! is_user_logged_in() || ! is_multisite() )
		return;

	// Show only when the user has at least one site, or they're a super admin.
	if ( count( $wp_admin_bar->user->blogs ) < 1 && ! is_super_admin() )
		return;

	if ( $wp_admin_bar->user->active_blog ) {
		$my_sites_url = get_admin_url( $wp_admin_bar->user->active_blog->blog_id, 'my-sites.php' );
	} else {
		$my_sites_url = admin_url( 'my-sites.php' );
	}

	$wp_admin_bar->add_menu( array(
		'id'    => 'my-sites',
		'title' => __( 'My Sites' ),
		'href'  => $my_sites_url,
	) );

	if ( is_super_admin() ) {
		$wp_admin_bar->add_group( array(
			'parent' => 'my-sites',
			'id'     => 'my-sites-super-admin',
		) );

		$wp_admin_bar->add_menu( array(
			'parent' => 'my-sites-super-admin',
			'id'     => 'network-admin',
			'title'  => __('Network Admin'),
			'href'   => network_admin_url(),
		) );

		$wp_admin_bar->add_menu( array(
			'parent' => 'network-admin',
			'id'     => 'network-admin-d',
			'title'  => __( 'Dashboard' ),
			'href'   => network_admin_url(),
		) );
		$wp_admin_bar->add_menu( array(
			'parent' => 'network-admin',
			'id'     => 'network-admin-s',
			'title'  => __( 'Sites' ),
			'href'   => network_admin_url( 'sites.php' ),
		) );
		$wp_admin_bar->add_menu( array(
			'parent' => 'network-admin',
			'id'     => 'network-admin-u',
			'title'  => __( 'Users' ),
			'href'   => network_admin_url( 'users.php' ),
		) );
		$wp_admin_bar->add_menu( array(
			'parent' => 'network-admin',
			'id'     => 'network-admin-t',
			'title'  => __( 'Themes' ),
			'href'   => network_admin_url( 'themes.php' ),
		) );
		$wp_admin_bar->add_menu( array(
			'parent' => 'network-admin',
			'id'     => 'network-admin-p',
			'title'  => __( 'Plugins' ),
			'href'   => network_admin_url( 'plugins.php' ),
		) );
		$wp_admin_bar->add_menu( array(
			'parent' => 'network-admin',
			'id'     => 'network-admin-o',
			'title'  => __( 'Settings' ),
			'href'   => network_admin_url( 'settings.php' ),
		) );
	}

	// Add site links
	$wp_admin_bar->add_group( array(
		'parent' => 'my-sites',
		'id'     => 'my-sites-list',
		'meta'   => array(
			'class' => is_super_admin() ? 'ab-sub-secondary' : '',
		),
	) );

	foreach ( (array) $wp_admin_bar->user->blogs as $blog ) {
		switch_to_blog( $blog->userblog_id );

		$blavatar = '<div class="blavatar"></div>';

		$blogname = $blog->blogname;

		if ( ! $blogname ) {
			$blogname = preg_replace( '#^(https?://)?(www.)?#', '', get_home_url() );
		}

		$menu_id    = 'blog-' . $blog->userblog_id;
		$site_title = $blavatar . $blogname;

		$wp_admin_bar->add_menu( array(
			'parent'    => 'my-sites-list',
			'id'        => $menu_id,

			// Adding a new filter here.
			'title'     => apply_filters(
				'ms_custom_site_menu_titles__admin_bar_network_site_title',
				$site_title,
				$blavatar,
				$blogname,
				$blog->userblog_id
			),
			'href'      => admin_url(),
		) );

		$wp_admin_bar->add_menu( array(
			'parent' => $menu_id,
			'id'     => $menu_id . '-d',
			'title'  => __( 'Dashboard' ),
			'href'   => admin_url(),
		) );

		if ( current_user_can( get_post_type_object( 'post' )->cap->create_posts ) ) {
			$wp_admin_bar->add_menu( array(
				'parent' => $menu_id,
				'id'     => $menu_id . '-n',
				'title'  => __( 'New Post' ),
				'href'   => admin_url( 'post-new.php' ),
			) );
		}

		if ( current_user_can( 'edit_posts' ) ) {
			$wp_admin_bar->add_menu( array(
				'parent' => $menu_id,
				'id'     => $menu_id . '-c',
				'title'  => __( 'Manage Comments' ),
				'href'   => admin_url( 'edit-comments.php' ),
			) );
		}

		$wp_admin_bar->add_menu( array(
			'parent' => $menu_id,
			'id'     => $menu_id . '-v',
			'title'  => __( 'Visit Site' ),
			'href'   => home_url( '/' ),
		) );

		restore_current_blog();
	}
}

Hint if you use MultilingualPress

I see in the first method the defining of language specific sites in the network. If you will pursue this with the MultilingualPress plugin, then is it a better alternative to filter the name of each site with the custom site name of each site.

• Overview of helpers to MLP: https://github.com/inpsyde/multilingual-press/wiki/Add--ons-and-Helper-Plugins
• MultilingualPress Alternative Title to Admin Bar Site links

@bueltge Thanks, very helpful!