The 49-year-old Frenchman was taken to Al-Jala hospital, where he died, after being hit by three bullets in an attack on the centre.
He was an employee of a company working on the extension and is reported to have been with colleagues at the time of the shooting.
Benghazi was the cradle of the Libyan uprising against Moamer Kadhafi and remains restive, with attacks on security and other targets nearly every day.
Islamist groups are now active and seven Egyptian Christians were executed there last week after being kidapped.
France has been heavily involved in renovation and extension work at the Benghazi medical centre since 2007 under a cooperation project.
A French ex-honorary consul to Benghazi, Jean Dufriche, headed France's team at the hospital but left Libya after an assassination attempt in July 2013.
In 2012 the US ambassador to Libya and three other Americans were killed in an attack on the US consulate there and an American teacher was killed in the city in December.
Two French guards were wounded by a car bomb outside the French embassy in Tripoli last April.