**Core Concept**
The patient's symptoms of headache and flushing, combined with a family history of a thyroid tumor, suggest a possible case of Multiple Endocrine Neoplasia Type 2 (MEN2), a hereditary condition characterized by the development of medullary thyroid carcinoma (MTC), pheochromocytomas, and hyperparathyroidism.

**Why the Correct Answer is Right**
The investigation required for this patient would be a **genetic test for RET proto-oncogene mutations**. The RET gene is a tumor suppressor gene that plays a critical role in the development and function of the thyroid gland, adrenal glands, and parathyroid glands. Mutations in the RET gene are associated with MEN2A and MEN2B, which are inherited in an autosomal dominant pattern. The presence of a RET mutation would confirm the diagnosis of MEN2 and guide further management, including surgical removal of the thyroid gland to prevent the development of MTC.

**Why Each Wrong Option is Incorrect**
**Option A:** A thyroid function test would be useful in evaluating hypothyroidism or hyperthyroidism, but it would not be the primary investigation for a patient with a suspected diagnosis of MEN2.

**Option B:** A parathyroid hormone (PTH) level would be useful in evaluating hyperparathyroidism, but it would not be the primary investigation for a patient with a suspected diagnosis of MEN2.

**Option C:** A 24-hour urine metanephrine test would be useful in evaluating pheochromocytomas, but it would not be the primary investigation for a patient with a suspected diagnosis of MEN2.

**Clinical Pearl / High-Yield Fact**
Patients with MEN2 are at increased risk of developing MTC, which can be asymptomatic until it has metastasized. Early detection and surgical removal of the thyroid gland can significantly improve outcomes for these patients.

**Correct Answer: A. Genetic test for RET proto-oncogene mutations.**