Can my mother sue the dialysis clinic?

What would you mother be suing for? For her arm being bruised under the skin and swollen? If that is, as we hope, the worst that happens to your mother, there is no point in suing--she'd spend more on the lawsuit than she'd get back. In a malpractice suit, all you can get back are additional medical costs caused by the malpractice; for long lasting (many weeks at a minimum) injuries which caused significant life impairment (a large impact on daily life) some amount for "pain and suffering," but that amount will be very small without months or longer of a serious impact, such as difficulty walking, eating, dressing yourself, etc.; and lost wages, if any, due to the malpractice. So with what you describe, there would be very little she could get back. Meanwhile, even without considering the cost of a lawyer--which you should have, since a malpractice suit is much more complicated than a small claims case--you must hire a medical professional to examine you, write a report, and testify in court about the cause of the injury, why it was malpractice, and the impact on your life. Such a medical professional (e.g. doctor) can easily cost many hundreds or several thousands of dollars. So unless something much worse happens to your mother, there is no point in thinking about a lawsuit.