Step 1: Interpret the thyroid function pattern. A high TSH with a low T4 indicates primary hypothyroidism. The thyroid gland is failing to produce thyroid hormone, and the pituitary responds by secreting more TSH to try to drive it, raising TSH.
Step 2: Localize the lesion. Because TSH is appropriately elevated, the pituitary is working normally; the defect is in the thyroid gland itself. This rules out pituitary and hypothalamic failure, where a low T4 would be accompanied by a low or inappropriately normal TSH.
Step 3: Identify the commonest cause. The most common cause of primary hypothyroidism in iodine sufficient areas is Hashimoto's (autoimmune chronic lymphocytic) thyroiditis.
Step 4: Exclude Graves disease. Graves disease causes hyperthyroidism with a low TSH and high T4, the opposite pattern.
Conclusion: The correct answer is option B, Hashimoto's disease.