While having a port installed for cancer treatments, the surgeon missed my mom's artery and punctured her lung deflating it. Does she have a case?

Not all bad outcomes are equivalent to malpractice.  Some risks are inherent and can occur under the best of care.  Further, even if there is malpractice, the damages have to be great enough to warrant the costs that will be incurred in bringing a malpractice action.  Those costs include retaining experts in the medical care involved that will be willing to testify against one of their colleagues.  Usually a collapsed lung is easily fixed by inserting a chest tube, and there are no long term health effects that would warrant a law suit.  Therefore, only if your mother experienced a dramatic outcome from the collapsed lung would there be a reasonable possibility of  recovery.