How do I display user name, role and site name using HTML tags inside a dashboard notification?

You can use the bloginfo( 'name' ) function to display the site name. Information about the current logged in user can be retrieved using the get_currentuserinfo() function.

Here is the fixed code:

function my_network_notice(){
    global $pagenow, $currentuser;
    get_currentuserinfo();
    if ( $pagenow == 'index.php') : ?>
        <div id="secondaryBox">
            <div id="author">
                <img src="https://wordpress.stackexchange.com/questions/77448/<?php echo get_stylesheet_directory_uri(); ?>/img/btn-articles-admin.png" width="40px" height="40px" />
                <h5><?php echo esc_html( $current_user->display_name ); ?></h5>
                <h6><?php echo esc_html( translate_user_role( $current_user->roles[0] ) ); ?>, <?php bloginfo( 'name' ); ?></h6>
            </div>
        </div>
    <?php endif;
}
add_action( 'admin_notices', 'my_network_notice' );

---

However, at the moment you’re just outputting the data straight to the dashboard. It would be better to create a dashboard widget (like the Right Now or Recent Drafts dashboard widgets):

/* display the dashboard widget output */
function wpse_77452_dashboard_widget_output() {
    global $currentuser;
    get_currentuserinfo(); ?>
    <div id="secondaryBox">
            <div id="author">
                <img src="https://wordpress.stackexchange.com/questions/77448/<?php echo get_stylesheet_directory_uri(); ?>/img/btn-articles-admin.png" width="40px" height="40px" />
                <h5><?php echo esc_html( $current_user->display_name ); ?></h5>
                <h6><?php echo esc_html( $current_user->roles[0] ); ?>, <?php bloginfo( 'name' ); ?></h6>
            </div>
        </div>
    <?php
} 


/* add the dashboard widget */
function wpse_77452_add_dashboard_widgets() {
    wp_add_dashboard_widget(
        'network_notice', // ID of widget
        'Important Information', // title of widget
        'wpse_77452_dashboard_widget_output' // function used to display widget
    );  
}

add_action( 'wp_dashboard_setup', 'add_notice_dashboard_widgets' );