The rough endoplasmic reticulum (option 3) is the correct answer because it is the site in the cell where proteins, including enzymes, are synthesized. The ribosomes attached on its surface give it a `rough` appearance, hence its name. These ribosomes are the machinery where protein synthesis occurs.

Option 1, the Golgi bodies, is incorrect. Golgi bodies are involved in modifying, sorting, and packaging the proteins for transport. They do not synthesize proteins.

Option 2 suggests that proteins are synthesized by the smooth endoplasmic reticulum. However, this is incorrect because the smooth endoplasmic reticulum is primarily involved in lipid metabolism and detoxification processes, but not in protein synthesis.

Option 4, lysosome itself, is also not correct. Although lysosomes contain enzymes, they do not have the machinery necessary to synthesize proteins. They serve as the digestive system of the cell, using their enzymes to break down and recycle a variety of materials.

Therefore, the rough endoplasmic reticulum is the correct answer as it is responsible for the synthesis of proteins, including the digestive enzymes found in the lysosomes.