How to change the name of the “edit my profile” link in the WordPress admin backend

You could use the wp_before_admin_bar_render hook. You can play with code samples on the Codex page by putting them in your theme’s functions.php.

First off you can try examining the object by outputting its contents with print_r(), and you’ll see all the familiar items. I would probably use remove_menu helper and then add a new one with the modified link title, just to keep things clean.

add_action( 'wp_before_admin_bar_render', 'my_tweaked_admin_bar' ); 

function my_tweaked_admin_bar() {
    global $wp_admin_bar;

    $wp_admin_bar->remove_menu('edit-profile');

    $wp_admin_bar->add_menu( array(
            'id'    => 'edit-profile',
            'title' => 'Edit Zee Profile',
            'href'  => admin_url('profile.php'),
            'parent'=>'user-actions'
        ));
}

The downside of this approach is that the new item will not show up in the same spot.