A synthetic organ was an organ of unnatural origin.
While talking about the mechanical components of the Borg drones in armory in 2153, Doctor Phlox asked Lieutenant Reed, "If your heart was damaged, would you want me to replace it with a synthetic organ, or would you rather die?" (ENT: "Regeneration")
Synthetic organs were sometimes used in battlefield situations, though Doctor Bashir commented how it was difficult knowing what organs to keep on hand. Bashir claimed that the Holy Grail of organ replacement would be to come up with a way to inject the patient with undifferentiated tissue that could become whatever organ was needed. (DS9: "When It Rains...")