Your neighbor is lucky to live next door to such a caring person. It is very kind of you to care about her well-being enough to try and find ways to help assist her when she is confused and frightened.
When these incidents occur try to maintain a calm and relaxed demeanor. If she sees that you’re calm and relaxed it may serve to decrease her anxiety. Speak softly but firmly reassure her that no one is there. You do not want to take a loud tone because it might frighten her but you also want to be authoritative. She may get angry with her son when he says that “there is nobody out there” because of his tone of voice. Perhaps the manner in which he speaks to her is dismissive or flippant. Also, continue to do as you have done, which is to walk to the yard with her to show her that no one is there and no damage has occurred. You may have to do this many times but if it helps her to feel better then it’s worth doing.
It might also be helpful to inquire about whether there is anyone she can call or have come over, other than the police, when these incidents occur. Is there another family member other than her son who she can call who has the ability to calm her? You can also suggest to her son that he take her to a doctor to have her evaluated or to have her medication adjusted. Paranoia can be a sign of Alzheimer’s. But it might also be a sign of an uncontrolled disorder that can be treated. I hope this helps.