**Core Concept**
The sympathetic nerve supply plays a crucial role in maintaining the tone of the eyelid, particularly through the superior tarsal muscle and Müller's muscle. This sympathetic innervation is responsible for the normal elevation of the eyelid.

**Why the Correct Answer is Right**
Damage to the sympathetic nerve supply disrupts the normal tone of the eyelid, leading to a characteristic drooping of the eyelid, also known as ptosis. This is because the sympathetic nerves stimulate the superior tarsal muscle and Müller's muscle to contract, thereby maintaining the normal elevation of the eyelid. The loss of sympathetic tone results in the eyelid drooping due to the unopposed action of the parasympathetic nerves, which are responsible for the constriction of the pupil and the lacrimation.

**Why Each Wrong Option is Incorrect**
**Option A:** Damage to the Edinger-Westphal nucleus primarily affects the parasympathetic innervation of the eye, leading to issues with pupillary constriction and accommodation, rather than eyelid elevation.

**Option B:** Damage to the motor part of the facial nerve would lead to weakness or paralysis of the orbicularis oculi muscle, resulting in difficulty closing the eyelid, but not necessarily causing eyelid drooping.

**Option D:** Damage to the lacrimal nerve would primarily affect the lacrimation, leading to dry eye or epiphora, rather than eyelid elevation or drooping.

**Clinical Pearl / High-Yield Fact**
It's essential to remember that Horner's syndrome, which is characterized by ptosis, miosis, and anhidrosis, is often the result of damage to the sympathetic nerve supply, particularly in the context of a carotid dissection or a thoracic outlet syndrome.

**✓ Correct Answer: C. Damage to sympathetic nerve supply**