My wife, after basically getting over a head cold, will then develop a hacking cough that may go on for sometimes two weeks or more.
She has had this happen, she says, since she was younger woman (now 60).
Does anybody have any ideas as to cause and/or best way to relieve the coughing.

This could be caused by post nasal drip due to the head cold. She should see her doctor to get a prescription to dry this up.

Take an antihistamine to dry up the nasal secretions. I get a lot of post nasal drip stiuations after a cold and this helps me otherwise I have a cough for weeks.

Coughs can often hang on for quite sometime after a cold, but as was said, good idea to have it checked, if she hasn't already, just to be sure there isn't another issue that developed.
Some people find hot tea with a bit of honey helps with a nagging cough. Or sucking on hard candy.

I had a cough for a whole month last year so I went to the doctor and got antibiotics. It went away in about three days. Your wife probably has chronic bronchitis, which means that she gets an upper respiratory infection almost every year. By brother and I both have it. A good way to relieve it is drinking hot water mixed with lemon juice and honey or just swallowing a spoon of warm honey. She could also try some herbal teas.