Step 1: A nerve that can be seen and is enlarged superficially over the neck must be a cutaneous nerve lying just under the skin, which immediately narrows the field to a cervical plexus branch.
Step 2: The greater auricular nerve fits perfectly. It is formed from $C2$ and $C3$ ventral rami, with the larger share from $C2$, and carries sensation from the ear, the skin over the parotid and mastoid, and the deep parotid fascia.
Step 3: Its course is distinctive - it wraps around the posterior edge of sternocleidomastoid at Erb's point and runs up across the muscle, making it the classic superficial neck nerve that is visible or palpable when thickened.
Step 4: Cranial nerves VII, X and IX run deep and cannot present as a visible subcutaneous cord, so they are ruled out, leaving the greater auricular nerve as the answer.
\[\boxed{\text{Greater auricular nerve}}\]