The only way to get a truely "open ticket" is to order an old fashioned paper ticket through a travel agency or purchase at a ticket counter. They are not very convenient or popular - full fare economy, offer no opportunity for upgrade and if you lose or tear the ticket you pay a fee to have it reprinted. They do not exist in the electronic world and phone reps cannot help with these reservations.

It sounds like you have a standard economy fare eticket. There are fees to change and the new ticket can be emailed to your family member. Just remember that all travel needs to be completed within one year if the original purchase.