Do patients have a right to have a family member present in the doctors examination room?

Yes. Assuming the patient is mentally competent to make
that decision for himself or herself the patient can request the presence of a family
member  or significant other during a physical exam. The HIPAA (Health Insurance
Portability and Accountability Act) covers a wide range of patient privacy issues, and
the federal government has issued numerous guidelines and clarifications regarding the
issue you are asking about here. (check out the links below to see the recommendations
as they are given to doctors).

For routine matters, the
patient's request to have the person in the room is seen as general permission to
discuss the patient's health or medical treatments.  The one big caveat is whether or
not to discuss something of a highly personal nature - HIV, for instance - with a
patient in front of the other person, whether they patient invited them or not. In such
a case, the doctor is advised to give the patient a heads-up as to what the discussion
may entail, and ask whether they want to continue, or ask the other person to step out
of the room first.