As long as you have given the name that's in her passport then it will be fine.
If the passport has a different name then you will have a problem as it will simply not match the name on the passenger list. Your sister more than likely wont make it as far as the check in desk if her name does not match one they have at the pre-check in security clearance. The staff there may take it up with the ticket desk, but then you're leaving it until 3 hours before departure to get a result. Make sure her ESTA is done in the passport name as well, or that's a whole other problem.
I do know some of the CO/UA staff at EDI, however I don't have a direct number to get in touch with them, sorry. I would advise going in to the terminal and speaking to someone on the ticket desk (near the international arrivals doors, between Menzies and EasyJet). The desk at EDI is only open from 6am until about 9:30am (after the flight leaves, it closes up) so keep that in mind. Whether you speak to someone from CO/UA or Menzies, you should get a good result by seeing them face to face.
I cannot say if they will charge a name change fee or not as I'm not sure of the airline policy for this kind of thing, but you'll get a lot further speaking with them at EDI than you will at a call centre.