drupal_mail() and names with apostrophes
I assume a few things: 


You have no exotic mail transport installed, having Drupal fallback to phps' mail()
You deliver the mail trough a unix system. Windows deals differently with mail.


Digging into RFC2822 you will find about escaping " and some other characters. 

The solution then presents itself as

$to = "\"Another O'Neil\" &lt;anotheroneil@example.com&gt;";
drupal_mail('contact', 'page_mail', $to, language_default(), $values, $from);

print $to; #=&gt; "Another O'Neil" &lt;anotheroneil@example.com&gt;