Mother has a cough and it's not going away. She had xrays and nothing was found. She is still coughing.?

More detailed work. First thing as a cardiologist i always like to make sure i'm not causing the problem. Medications like ace inhibitors can cause chronic cough. Always like to rule out congestive heart failure and asthma too. A good history and physical , ekg , echocardiogram , pulmonary function study and an ambulatory pulse oximeter to start. A bnp level in the blood can be checked if high is suggestive of chf.
Possibilities... Possibilities include bronchospasm from any etiology, including COPD especially if she has ever smoked. If she smokes now, she needs to quit to decrease airway inflammation. Other possibilities include sinus disorders, gerd or a combination of these factors. Treatment depends on etiology so she needs to see her doctor for evaluation.
Chronic cough. I agree with the above answer but xray are not always the best, a low dose ct scan could be preformed to rule out other possibilities. Sometimes mile interstitial lung disease or mai can be seen on ct but not xray.