Correct answer: Hashimoto's thyroiditis.
Step 1: The clue cluster is a young woman, an elevated TSH (so she is hypothyroid), and a biopsy showing lymphocytic infiltration with Hurthle (oxyphil) cells. This is the classic picture of Hashimoto thyroiditis, the most common cause of hypothyroidism in iodine-sufficient areas.
Step 2: Autoimmune destruction of the gland causes gradual thyroid failure, so TSH rises. Histology shows widespread mononuclear infiltrate with small lymphocytes, plasma cells, and well-developed germinal centres. Follicles atrophy and are lined by Hurthle cells with abundant eosinophilic granular cytoplasm.
Step 3: Why not the others. Graves disease causes hyperthyroidism with low TSH, not elevated TSH. Follicular carcinoma shows capsular and vascular invasion, not lymphocytic infiltrate. Medullary carcinoma arises from C cells and shows amyloid, not Hurthle cells.
Ref: Robbins, endocrine system, thyroiditis.