wordpress bar

Not sure if this covers everything you need, but it might be a good start to cover customization!

How to Customize the WordPress Admin Area
How to Customize the WordPress Admin Easily

A quick google search brought up this:

Drop this in your functions.php file

function replace_howdy( $wp_admin_bar ) {
    $my_account=$wp_admin_bar->get_node('my-account');
    $newtitle = str_replace( '<my style="background-color: rgb(75, 254, 120); border: 1px solid black; border-top-left-radius: 3px; border-top-right-radius: 3px; border-bottom-right-radius: 3px; border-bottom-left-radius: 3px; font-weight: bold;">Howdy</my>,', 'Logged in as', $my_account->title );
    $wp_admin_bar->add_node( array(
        'id' => 'my-account',
        'title' => $newtitle,
    ) );
}
add_filter( 'admin_bar_menu', 'replace_howdy',25 );

There is also:
http://wpteach.com/change-the-howdy-in-wordpress-admin-bar/

And check the plugins section, I know there is at least two that will customize/remove the ‘Howdy’ notification all together.

Oh wait, are you talking about the admin bar itself? This is only visible to you when you are logged into your site. It allows you to log in and out, return to your dashboard, add new posts, etcetera. Your readers can’t see it. (unless they’re logged in, and then it’s their name)