Change the contents of a dropdown through the admin panel?

I think this is really what taxonomies are for, so if a custom taxonomy won’t work, you could always still create the custom taxonomy and populate the drop down with the taxonomy. This would allow the user to create new terms via the taxonomy. There is a plugin called Meta Box, and it makes stuff like this a breeze!

Update
After commenting with you back and forth below, I think what you are really looking for is just a custom taxonomy that will allow the user to add, subtract, and apply terms to a post or page. A taxonomy is one of those boxes on the right side of the post or page creation screen like “tags” or “categories” that allows a user to add terms to a post, which allow the posts to be searchable, more organized and much more filterable.

If you want you can just copy paste this into your functions.php file to create a new taxonomy and append it to the posts and pages admin screen. This is straight from the WP documentation.

// hook into the init action and call create_book_taxonomies when it fires
add_action( 'init', 'create_places_taxonomies', 0 );

// create two taxonomies, places and writers for the post type "book"
function create_places_taxonomies() {
    // Add new taxonomy, make it hierarchical (like categories)
    $labels = array(
        'name'              => _x( 'Places', 'taxonomy general name' ),
        'singular_name'     => _x( 'Place', 'taxonomy singular name' ),
        'search_items'      => __( 'Search Places' ),
        'all_items'         => __( 'All Places' ),
        'parent_item'       => __( 'Parent Place' ),
        'parent_item_colon' => __( 'Parent Place:' ),
        'edit_item'         => __( 'Edit Place' ),
        'update_item'       => __( 'Update Place' ),
        'add_new_item'      => __( 'Add New Place' ),
        'new_item_name'     => __( 'New Place Name' ),
        'menu_name'         => __( 'Place' ),
    );

    $args = array(
        'hierarchical'      => true,
        'labels'            => $labels,
        'show_ui'           => true,
        'show_admin_column' => true,
        'query_var'         => true,
        'rewrite'           => array( 'slug' => 'places' ),
    );

    register_taxonomy( 'places', array( 'post', 'page' ), $args );
}

If you would like to change the functionality to a more “tag” like function, then simply changed the hierarchical setting to false.

'hierarchical'      => true,

You really should be using taxonomies to perform this function rather than a custom drop down box. Should your client want to expand upon the functionality, this is going to make it very easy to do so in the future.