Step 1: Two compartments to remember. The cavernous sinus has structures that truly traverse its lumen and others that are merely held in its lateral wall. Distinguishing the two is the key to this question.
Step 2: Inside the cavity. Only the internal carotid artery and the abducent nerve travel within the venous channel of the sinus. Among the four choices, the internal carotid artery is the one that fits this description.
Step 3: Rule out the rest. The maxillary division of the trigeminal nerve sits in the lateral wall rather than the lumen. The mandibular division never enters the sinus and exits via the foramen ovale. The facial nerve takes an entirely separate course through the temporal bone and is not a content of the sinus.
Step 4: Decision. The artery that courses through the medial part of the sinus is the internal carotid artery.
\[\boxed{\text{Internal carotid artery}}\]