What is it that causes an elderly person's hands to shake?

Hand tremble. The most common reason is "benign essential tremor". It usually happens as you get older and can worsen as you get older. There is a strong genetic disposition. It occurs most when a person is "intentionally" doing something like writing or pointing. It can be helped by medicines such as "b-blockers". Of course, always talk to your doctor about your symptoms to make a definitive diagnosis.
PD and ET common. Parkinson's disease and essential tremors are both common in elderly. Tremors of pd occur at rest as opposed to et which occur with action. Other less common causes are some medications, alcohol withdrawal, hyperthyroidism and anxiety.