Can a doctor prescribe synthroid (thyroxine) to increase a patient's appetite?

Not wise: That would not be a good idea. A high dose of synthroid (thyroxine) would increase metabolic rate (and put a stress on the heart) and appetite may/may not go up, but usually it would cause weight loss. Synthroid (thyroxine) should only be prescribed for hypothyroidism, or other clear thyroid related conditions, and should not be used for obesity or appetite.