The name of the organism growing in a woman's uterus during pregnancy is the fetus. The fetus is the stage of prenatal development after the embryo and before birth. It is the developing human offspring from the end of the eighth week after fertilization until birth. During this stage, the fetus undergoes significant growth and development, with all major organs forming and functioning by the end of the first trimester. The fetus continues to grow and mature throughout the pregnancy until it is ready to be born. Therefore, the correct answer to the question is:
A. fetus
This is the organism that grows in a woman's uterus during pregnancy.