**Core Concept**
Pancreatic islet cell tumors are neoplasms arising from the endocrine component of the pancreas, specifically from the islet cells that produce various hormones. These tumors can be functioning or non-functioning, and the functioning tumors lead to excessive hormone production, causing specific clinical syndromes.

**Why the Correct Answer is Right**
The most common functioning pancreatic islet cell tumor is the insulinoma. Insulinomas are characterized by the excessive production of insulin, leading to hypoglycemia. This is due to the unregulated secretion of insulin from the tumor cells, which can be caused by mutations in the MEN1 gene, leading to the activation of the insulin gene promoter. Insulinomas often present with symptoms of hypoglycemia, such as confusion, tremors, and sweating.

**Why Each Wrong Option is Incorrect**
**Option A:** Glucagonoma is a rare functioning pancreatic islet cell tumor that produces excessive glucagon, leading to hyperglycemia. This is in contrast to insulinoma, which causes hypoglycemia.
**Option B:** Somatostatinoma is a rare functioning pancreatic islet cell tumor that produces excessive somatostatin, leading to diabetes mellitus and malabsorption. While it is a functioning tumor, it is not the most common type.
**Option C:** Gastrinoma is a functioning pancreatic islet cell tumor that produces excessive gastrin, leading to peptic ulcer disease. While it is a common type of functioning tumor, it is not the most common.

**Clinical Pearl / High-Yield Fact**
Insulinomas are often detected incidentally on imaging studies or during evaluation for symptoms of hypoglycemia. They can be challenging to diagnose due to their small size and location, but a 72-hour fasting test can help confirm the diagnosis.

**Correct Answer:** A. Insulinoma