Hey
OSTicket 1.6ST shows who closes the ticket up where internal notes are posted. You have to be viewing the ticket and you'll see:
"Insert_Name closed the ticket on reply" or "Ticket closed without response by Insert_Name"
If you're wanting to see the name of the person who closed the ticket in the Ticket List Page, where it list all the closed tickets, you'll have to modify one of the other mods that show how to insert new columns into that page. Like what was done with the (Last Update Column) or the (Assign Staff Column).
Hope this helps.
G.Dawkins