**Core Concept**
The patient's symptoms of weakness, vomiting, hypotension, and skin and mucous membrane pigmentation are suggestive of a condition affecting the autonomic nervous system, specifically the adrenal glands. The pigmentation is due to the increased production of melanocyte-stimulating hormone (MSH) by the pituitary gland in response to low blood pressure.

**Why the Correct Answer is Right**
The correct diagnosis is Addison's disease, a rare endocrine disorder caused by the failure of the adrenal glands to produce sufficient cortisol and aldosterone. This leads to a compensatory increase in the production of MSH, resulting in skin and mucous membrane pigmentation. The patient's symptoms are due to the lack of cortisol, which is essential for maintaining blood pressure. The pituitary gland attempts to compensate by releasing more adrenocorticotropic hormone (ACTH), which stimulates the adrenal glands to produce more cortisol, but this is insufficient.

**Why Each Wrong Option is Incorrect**
* **Option A:** Serum cortisol levels would not be diagnostic in this case, as they can fluctuate and may not accurately reflect the patient's condition.
* **Option B:** Serum electrolyte levels may show hypokalemia and hyponatremia, but these findings are not specific for Addison's disease.
* **Option C:** A 24-hour urine collection for 17-ketosteroids may be used to diagnose adrenal insufficiency, but it is not the most definitive test.

**Clinical Pearl / High-Yield Fact**
Addison's disease is often associated with a characteristic hyperpigmentation of the skin and mucous membranes, which is due to the increased production of MSH. This can be a key diagnostic clue in patients presenting with weakness, vomiting, and hypotension.

**Correct Answer: C. 24-hour urine collection for 17-ketosteroids.**