Extending the search function to a specific meta field value in the admin posts table

You can use the meta_query parameter of WP_Query to search custom fields.

Untested:

$meta_query = $query->get( 'meta_query', array() );

$meta_query[] = array(
    'key'     => 'function_name',
    'value'   => $query->query_vars['s'],
    'compare' => 'LIKE',
);

$query->set( 'meta_query', $meta_query );