The hormone that primarily promotes the production of red blood cells in the bone marrow is erythropoietin. Erythropoietin is a glycoprotein hormone mainly produced by the kidneys in response to low oxygen levels in the blood, a condition known as hypoxia. It acts by stimulating the proliferation and differentiation of erythroid progenitor cells in the bone marrow, leading to increased red blood cell (RBC) production. This process is crucial for maintaining adequate oxygen delivery to tissues throughout the body.

In contrast, thyroxine is involved mainly in regulating metabolism, growth hormone promotes overall body growth and development, and cortisol is a glucocorticoid that helps manage stress and inflammation but does not directly stimulate RBC production.

Therefore, erythropoietin is the hormone specifically responsible for red blood cell synthesis.

Reference: Guyton and Hall Textbook of Medical Physiology, 13th Edition, Chapter 16 / Page 190