Iodine is good, but you need to watch out for the
Wolff-Chaikoff effect. Eating a reasonable amount of iodized salt and/or fish is good, taking loads of iodine might not be.
With regard to coconut oil: I hadn't heard of this, so I googled it. The Mayo Clinic is beyond skeptical, offering a flat no:
http://www.mayoclinic.com/health/coconut-oil-thyroid/AN01367/
I figured it was worth looking into anyway.
Rabbits fed coconut oil had lower TSH levels and gained less weight compared to rabbits fed soybean oil. The authors state this:
The consumption of saturated fats has shown to assist in regulating the healthy functioning of the thyroid gland and combating the hypothyroid like symptoms such as fatigue, weight gain, water retention that may arise through a modern diet high in polyunsaturated fats.(6)
But the citation is bunk. The study is good, but I'm not yet convinced. There are a number of claims -- you'll find them online -- that polyunsaturated fats are bad for thyroid function, and most claimants recommend substituting with saturated fats. Unfortunately, the layer of documented empirical evidence that floats to the top of this Internet bullshit slurry is rather thin, but it goes hand-in-hand with better, more reliable evidence regarding other bodily functions that seems to recommend a lowered polyunsaturated fatty acid intake.
And then there's Mark's Daily Apple. I like Mark, he knows his shit and checks his sources.
http://www.marksdailyapple.com/hypothyroid/#axzz21tikCvN2
Mark links us to
an extensive compilation of resources re: thyroid disorders. Among the recommendations is selenium, a mineral which is incredibly hard to pin down in foods -- the soil food is grown in can result in the same food having several times as much selenium in the United States as in selenium-deficient soils in China. So selenium is the odd mineral you might actually want to consider supplementing, and the pills are really cheap since you only need a tiny amount.
And coconut oil? While its impact on thyroid function is uncertain, it is known to
help with weight maintenance and seems to
increase thermogenesis and thus the basal metabolic rate. It may alleviate some of the symptoms of hypothyroidism, though its impact on the underlying problem is unclear.
Goat cheese and butter contain some of the same fats as coconut oil. If you don't like the taste of coconut, they're a viable (if expensive) option.