My mother's hand has started to shake while holding a cup, etc. Am I am responsible because of stress in our family?

Many things can cause tremors in the hands. It could be medications she's taking or the beginning of a neurological disorder. Or stress. Or any number of things. She really should see a doctor. I hope you see what you can do about family stress, too, for everyone's sake.
Take care,
Carol

My mother has Parkinson's Disease and her first symptoms were occasional shakiness and forgetfulness. Schedule an appointment with her physician and go with her to relay symptoms you've observed.