**Core Concept**
The question is testing the diagnosis of a rare genetic disorder characterized by recurrent episodes of severe headache, ataxia, and vertigo, with a positive family history.

**Why the Correct Answer is Right**
The most likely diagnosis is Familial Hemiplegic Migraine (FHM). FHM is a subtype of migraine with aura, which is a neurological disorder that can cause severe headache, ataxia, vertigo, and other neurological symptoms. The condition is inherited in an autosomal dominant pattern, meaning that a single copy of the mutated gene is sufficient to cause the condition. The pathophysiology of FHM involves mutations in genes that encode for subunits of voltage-gated calcium channels, such as CACNA1A and SCN1A, which can lead to abnormal neuronal excitability and neurotransmitter release.

**Why Each Wrong Option is Incorrect**
* **Option A:** Tension headache is a common type of headache disorder, but it is not typically associated with ataxia and vertigo, and it does not have a strong genetic component.
* **Option B:** Cluster headache is a severe form of headache disorder, but it is not typically associated with ataxia and vertigo, and it does not have a strong genetic component.
* **Option C:** Stroke is a medical emergency that can cause sudden onset of neurological symptoms, including headache, ataxia, and vertigo. However, it is not typically associated with a positive family history of similar headaches.

**Clinical Pearl / High-Yield Fact**
FHM can be differentiated from other types of migraines by the presence of ataxia and vertigo, as well as a positive family history of similar headaches.

**Correct Answer:** C.