Read your travel insurance. Most will offer coverage for certain family members if the illness did not exist when the flight was booked (or was declared) and could not be predicted.
Contact your health insurance company. Some of the more expensive policies may cover the cost of canceled holidays if it is due to a medical condition. You may need to claim off the insurance of the person who is ill.
Contact your airline. If you bought a nonrefundable flight, you may find that the airline is unwilling to do anything. However, most airlines will allow you to move the flight to use on a different date, although you may have to pay a fee to change the date, and any difference in flight costs between the old and new date.
Discuss the family member's illness with a doctor, and see what the prognosis is. Try to establish an expected release date, if possible.
Evaluate the costs of canceling the flight. Depending on the illness, the cost of the flight and the reason for traveling, you may be better off to take the flight and book an earlier one back, or simply be available if you are needed.