How can one deal with a mother who is having an illicit relationship with a strange man over the phone and refuses to stop?

The mere fact that your mother is speaking with a non-mahram man does not, by itself, indicate an illicit relationship. The default is to have a good presumption [of others]. If, however, an unlawful relationship is proven, then you must supplicate for her, as Allah Almighty says: "Call upon Me; I will respond to you." [Ghafir: 60] And continue to advise her gently and kindly, explaining to her the gravity of her conduct. If she responds, then praise be to Allah. If she persists, then work to prevent her from the causes of corruption, while remembering that she is your mother. In parallel, contact the man and threaten to report him to the authorities if he repeats such behavior.