How can I change the admin search posts fields?

It is possible, but you’ll have to play a little bit with the actual query. As always, the furious posts_clauses filter comes to action:

function wpse_alter_posts_search( $pieces )
{
    global $wpdb;

    // Make the input save
    $search_string = like_escape( $_GET['s'] );

    // Your new WHERE clause
    $where  = $wpdb->prepare(
        "$wpdb->postmeta.%s LIKE %s",
        'YOUR_COLUMN', // Should match your col name 1)
        "%{$search_string}%"
    );

    // Not sure if this exactly the same on your install 2)
    $pieces['where'] = str_replace( 'AND (((', "AND ((({$where} ", $pieces['where'] );

    return $pieces;
}
add_filter( 'posts_clauses', 'wpse_alter_posts_search' );

NOTES

1. Could be that you need to unserialize some parts to be search able.
2. You better simply dump the $pieces to take a look into them, before altering the query (right after the $search_string was received).