How to show a custom notification to a specific user?

You can show admin notices via the admin_notices action. You can display it for a specific user by checking the contents of the global $current_user:

function wpa80367_admin_notice() {
    global $current_user;
    if ( 'someusername' == $current_user->user_login ):
        echo '<div class="updated"><p>';
        echo 'This is an admin notice just for you, someusername.';
        echo '</p></div>';
    endif;
}
add_action( 'admin_notices', 'wpa80367_admin_notice' );

You could also extend this by saving per-user notices as user metadata via add_user_meta, then checking if a user has a notice in the above function via get_user_meta.

Another option is to display messages in a Dashboard Widget rather than admin notice.