How to force listing contributors in authors dropdown menu

Here’s a solution that will remove the original author meta box and replace it with a similar, but customized version which includes users with the contributor role.

The logic for adding/removing the author meta box is pulled directly from the core. The meta box display callback is also cloned from the core, but we use the role__in parameter of wp_dropdown_users() which lets us specify which roles we want to include within the dropdown.

/**
 * Removes the original author meta box and replaces it
 * with a customized version.
 */
add_action( 'add_meta_boxes', 'wpse_replace_post_author_meta_box' );
function wpse_replace_post_author_meta_box() {
    $post_type = get_post_type();
    $post_type_object = get_post_type_object( $post_type );

    if ( post_type_supports( $post_type, 'author' ) ) {
        if ( is_super_admin() || current_user_can( $post_type_object->cap->edit_others_posts ) ) {
            remove_meta_box( 'authordiv', $post_type, 'core' );
            add_meta_box( 'authordiv', __( 'Author', 'text-domain' ), 'wpse_post_author_meta_box', null, 'normal' );
        }
    }
}

/**
 * Display form field with list of authors.
 * Modified version of post_author_meta_box().
 *
 * @global int $user_ID
 *
 * @param object $post
 */
function wpse_post_author_meta_box( $post ) {
    global $user_ID;
?>
<label class="screen-reader-text" for="post_author_override"><?php _e( 'Author', 'text-domain' ); ?></label>
<?php
    wp_dropdown_users( array(
        'role__in' => [ 'administrator', 'author', 'contributor' ], // Add desired roles here.
        'name' => 'post_author_override',
        'selected' => empty( $post->ID ) ? $user_ID : $post->post_author,
        'include_selected' => true,
        'show' => 'display_name_with_login',
    ) );
}