During periods of extreme stress a client may experience elevated blood pressure, dilated pupils, and increased respirations. these unconscious responses originate in which part of the brain?

The answer is hypothalamus.

The hypothalamus regulates the sympathetic nervous system, part of autonomic nervous system, associated with fight-or-flight responses.
During stress situations the sympathetic nervous system is more activated, causing a response of either run or fight. Because of that, the body needs to be prepared and there will be elevated blood pressure (the blood is going more to the skeletal muscle, heart,etc), dilated pupils (for better peripheral vision) and increased respiration.