Final answer:
The axillary vein changes its name to the subclavian vein at the lateral margin of the first rib, continuing the blood's return journey to the heart.
Step-by-step explanation:
The vein that changes its name to the subclavian vein at the lateral margin of the first rib is the axillary vein. The axillary vein is a major vein in the axillary region that drains the upper limb. As it passes through the body wall and enters the thorax, it becomes the subclavian vein. This transition marks a continuation of the pathway for blood returning to the heart, as the subclavian vein ultimately leads to the brachiocephalic vein, which in turn contributes to the superior vena cava, returning blood to the heart.