It's not usually possible to change the name on a booking once tickets have been issued, but you may still be able to travel on the ticket. The deciding factor is whether your tickets are 'nominative' or not.
Nominative: Tickets with names on them can only be used by the named traveller, who must carry ID. They cannot be used by anyone else. To make a name change you'll need to refund your ticket and buy a new one with the correct name. There are some exceptions for Italo trains, see below.
Non-nominative: Anyone can travel on non-nominative tickets as long as they are permitted to travel on the same fare type e.g. an adult, senior or child can travel on an adult ticket, but an adult cannot travel on a child or senior ticket.
Operator Nominative / Non-nominative Name change allowed? Can someone else use my ticket?
*Not all Italo tickets permit name changes. You should check the specific fare conditions associated with your ticket.
What if I made a typo?
Variations such as abbreviated names, typos or spelling mistakes aren’t usually a problem. The ticket inspector (if they notice) will recognise that it's simply human error. Just make sure you carry ID.