I have healed my own hypothyroidism naturally, primarily through a healthy diet.
For the classic thyroid problem, goiter, a small does of iodine on a steady basis is the cure. That is rare in modern society these days, as we use iodized salt. If you think iodine supplementation will help, your best food choice is seaweed, usually called "sea vegetables," eaten cooked or raw, but minimally processed. (The thin nori sheets on sushi are not enough.) These provide iodine.
For other food choices, one has to consider that thyroid function is heavily dependent on the insulin/glucagon hormonal balance, and that thyroid troubles are often a symptom of an underlying condition, particularly a yeast or fungal infection. For more information on a diet that helps, see my hub "Is there a right way to eat?"