Thanks Craig and kmonroe!
I tested both ways and the one that worked was: <{$ticket[fullname]}>
(I'm not using HTML formated email tickets)
On a related note, I just realized that the new ticket alerts
already showed the name of the client that started the ticket. It shows it in the email FROM field (when you're looking at the email in Outlook or Eudora or whatever mail program you use

)
I think in our testing, we always used our own email addresses, so our names showed up there (and so we were both the client and the support person).