The word itself arises from the Greek. The root term [-phleb-] derives from [φλέβα] (phleba) meaning "vein", and the suffix [-otomy], meaning "to cut" or "to open". Let's not forget that the suffix component [-y] means "process of". So [phlebotomy] is the "process (or action) of cutting open a vein"
For centuries a standard practice in medicine was to "bleed" a patient, by opening a vein under controlled conditions and letting some blood flow. The practice was known as "bloodletting" or phlebotomy. Not in use today, it is said that excessive bloodletting contributed to the death of George Washington, having removed 5 pints of blood in one day!. Today the professionals who draw blood are called "phlebotomists"
The image (circa 1860) depicts one of the only known three photographs of a bloodletting procedure. Observe the lack of aseptic technique.