Your mother sounds like she has a problem called radiculopathy. This is caused by a nerve that is being compressed somewhere in her body. If the pain starts in her
shoulder, and goes all the way down to her hands, then I would think that her neck or her shoulder could be her problem. But, to complicate matters, she may have carpal tunnel too, as her hand symptoms sound quite severe.
If this was my patient, I would get xrays of the cervical spine as well as the affected shoulder. This would at least let me know if she has arthritis or some other kind of problem that would cause these symptoms. It may also help for her to see an orthopedist, as they may be able to pinpoint where her pain is coming from. The orthopedist may need to get an MRI of the neck or shoulder to find out in greater detail what is causing the problem so that a treatment plan can be implemented.
One other thought that went through my mind when I read this letter is that Physical Therapy may help your mother. They often can help reduce pain with gentle exercises, and they may offer more insight on what may be causing your mother's pain. Please ask your mother's primary care provider for a referral. Good luck!