Thank you to all who replied, unfortunately, I thought my initial post was clear enough as to my predicament, although it appears not.
So for the benefit of clarity I will try and simplify.
A ticket has been booked in one name, lets say Mrs Jane Smith. However, the name is actually Mrs Jane Jones. The passport is also in the name of Mrs Jane Jones. All other details, DOB/Address/Destination, times etc for the booking/eticket are correct.
After discussions with CO call centre, they say they cannot change 'major' name changes. Slight spelling, title, initial etc mistakes they can rectify.
It is not as Hartington puts it, anything to do with money. They will not do it, full stop!!! If money could change the details, then thats what would be done!!
CO call centre admitted that the advance passenger information (API) is correct (Mrs Jane Jones) but her eticket is Mrs Jane Smith. CO call centre says that US immigration do not use, or rely, on ticketing information, they use and rely on this API and ESTA.
As you say ExXB, in the many times I have visited the USA, I have never been asked to produce a return ticket.
What makes matters worse, and confusing, is CO call centre said that my own DOB was wrong (same as my sisters DOB) Thankfully this error is something they can change, but they admitted that my DOB was correct at point of booking ticket, but 'mysteriously' changed sometime in the last 4 months!!!
As I said in my initial post, CO said that a marriage certificate (Or old passport showing maiden name) would be sufficient to ensure admittance to the flight.
I had already decided to travel to EDI at the weekend to speak with the staff at the CO checkin desk - my reason for making a post here was to see if anyone knew [B]facts[B] on the matter,perhaps even something similar that has happened to themselves or knowledge, rather than speculation, which I can do myself.
However, I appreciate all constructive feedback on this case and hope to hear more if any available.