Think first about what a brown tumor actually is. It is not a true neoplasm. It is a reactive mass of giant cells and fibrous tissue that grows where bone has been eaten away by too much parathyroid hormone. The brown shade comes from old blood breakdown product, hemosiderin, trapped inside.
Now match the cause. You need a state of high PTH activity. Among the four choices, only hyperparathyroidism gives that. The high PTH switches on osteoclasts, bone is resorbed, and the empty pockets fill with fibrovascular tissue, producing osteitis fibrosa cystica and brown tumors.
The other three fail this test. Hyperthyroidism raises overall bone turnover and may thin bone, but never makes a brown tumor. Hypoparathyroidism means low PTH, so resorption is reduced. Hypothyroidism slows metabolism and bone turnover. None of these create the resorption pockets a brown tumor needs.
Supporting clues for hyperparathyroidism are high calcium, low phosphate, subperiosteal erosion of phalanges, and a salt-and-pepper skull on X-ray.
\[\boxed{\text{Hyperparathyroidism}}\]