My grandfather had a quadruple bypass several months ago and is still coughing and as an allied health professional I see this often in the hospital particularly the cardiovascular recovery unit. So, this coughing is common with this type of procedure. I know that doesn't give you must comfort. She and her doctors might just have to do their best to manage the situation. But, there is a good chance this can continue for months unfortunately.
My Grandfather was coughing so much that he had to miss an important family/religious event. It's really sad when you have a condition that restricts someone so much, and he was never like this his whole life. But, then again without the bypass your Mother and my Grandfather probably wouldn't be around at all.
You might want to try fresh whole Eucalyptus leaves with your mom. Soak it it hot water and have her breath it in--you can also make a tea with it. Just keep working with the docs until you find a long-term solution. But, realize that some days are going to be worse than others. Like I said, it's a management situation. Good luck with her! One other thing-- if her coughs are productive encourage her to get the mucous out and not let it settle back. If and when she can get up and walk that might help to get the blood circulating. When she has permission encourage her to walk.
She needs to find out why she is coughing. Have her call the cardiologist who is treating her. HOpe all is well.
Is she on an Ace Inhibitor medication such as Lisinopril, Enalapril, Captopril, or other "pril"? If so, is the cough a dry nonproductive type cough?