Where does most new membrane synthesis take place in a eukaryotic cell?

The correct answer is: endoplasmatic reticulum (ER).

Membranes are continuously renewing their molecules that are synthesized in endoplasmic reticulum and transported to their final location by vesicular trafficking (exocytosis of vesicles).

Cholesterol and glycerophospholipids  for the cell membranse are all synthesized in the endoplasmic reticulum because it contains enzymes involved in lipid synthesis. Proteins are inserted into the ER while they are still being synthesized and their synthesis is finished in the ER.