I booked a round-trip flight for me and my wife from Egypt to Los Angeles for a family visit. I had to stay longer in LA than previously planned. However my wife should get back PLUS my mother-in-law.
Is there any way my mother-in-law can replace me on the return flight?
There is no regulation that requires an airline to allow you change the name on a half-used international ticket anywhere I know of. So, whether or not you can endorse the ticket over or transfer it to your mother-in-law's name is governed primarily by the airline's Conditions of Carriage (or Contract of Carriage) and by the rules of the fare, both of which you agreed to when you purchased the ticket.
But as in the case of Can my travel companion use my ticket for the extra seat/space?, Using no-show ticket for infant, and What can I do with a plane ticket that won't be used?— in the overwhelming majority of cases— airline tickets are not transferable. For a random example, Air France states among its general provisions that
3.1.c A Ticket may not be transferred, subject to the applicable regulations in force, in particular relating to package holidays. If a person other than the person who is to travel presents a Ticket for carriage or refund purposes, the Carrier shall not assume any liability if, while acting in good faith, it carries or refunds the person who presents the Ticket.
Egyptair's language is substantially similar:
3.1.2 A Ticket is not transferable: 3.1.2 (a) If a ticket is presented by someone other than the person entitled to be carried there under or to a refund in connection therewith, carrier shall not be liable to the person so entitled if in good faith it provides carriage or makes a refund to the person presenting the ticket. 3.1.2 (b) Carrier reserves the right to request a Passenger to identify himself.
Cathay Pacific is more succinct:
3.1.2 You cannot transfer your Ticket.
You might be able to get away with it in the rather unlikely event that you and your mother-in-law share the same name; see Can two persons sharing a name travel one-way each on a return ticket?.
Allowing names to be changed would negatively impact the airline's revenue model, which is designed to extract more money from people who book travel at the last minute, who are mostly price-insensitive business travelers. Therefore, the airline will demand that you purchase a new ticket for your mother-in-law (or change her existing one, if any), and to extend your stay you will similarly need to change your ticket, subject to any fees and restrictions, or abandon it and buy a new one. Depending on the reason you are staying, travel insurance may be able to help with the latter.
It doesn't hurt to ask, of course, but considering you have already used the outbound segments, I doubt you will find an agent who is able and willing to change the name on your ticket to your mother-in-law's.
In general, no. You'd have to check the specific terms and conditions for your ticket or ask the airline, but most airline tickets are non-transferable (doubly so after you've flown on them), and the name on the ticket must match your ID.
If the circumstances involve something like a medical or family emergency, the airline may make an accommodation for you, such as waiving a change fee, as a customer service gesture. Otherwise, you will likely have to pay to change your ticket (or buy a new one should that wind up being cheaper) and purchase a separate ticket for your mother-in-law.
Like other answers, it is in general difficult, and possibly with some extra cost.
It is much better (in my opinion) that you try to change your ticket return dates. Your wife will flight back on original flight, your mother-in-law could buy a ticket on the same flight, and you will return with your original ticket, just on a different date.
Usually, plane tickets are non-transferrable. So your mother-in-law won't be able to use your ticket. You'll need to buy her a new one.
