Avoiding excess iodine makes sense (like eggs, fish and iodised salt, dairy) for a hyperthyroid, but you do need still need some I beleive.
Fresh water fish has less that seafood. If you can find some freshwater farmed salmon at some point, to eat occasionally this may be ideal because omega-3 is useful to support the thyroid, and though it still has some iodine, its much less that sea fish.
Being that they cant have dairy, eggs, or seafood on a lower iodine diet, I would suggest that grassfed land mammals is the most ideal for of protein and fat, being lower in iodine, and containing balanced fats and whole proteins.
And while we are on that topic, why cant they have meat? I understand avoiding cured meat, perhaps even avoid feedlot meat (often they put iodine in the feed), but grassfed cuts from the butcher?
Even with feedlot meat it tends to be lower than seafood:
In mcg/kg: Beef mean iodine levels in mcg/kg: Muscle (173), Liver (70), Kidney (61). Poultry - Breast/Muscle (56 – 1248)
(Tending towards to lower end usually, but dependant on iodine in the feed)
Versus mean iodine in mcg/kg: Cod (1100), Haddock (2500), Atlantic Salmon (760), Rainbow Trout (130)
As you can see, meat, even feedlot, is generally lower than seafood, depending on how over the top the farmers have gone with the iodine in the feed. And the more freshwater the fish is, the lower it is too. So freshwater fish and grassfed meats are fine...
Ergo best protein source: 100% Grassfed beef, mutton, buffalo and lamb (not minced as that can very rarely be contaimanated with thyroid gland), and occasional 100% freshwater farmed salmon or freshwater trout (wild salmon tends to roam into the sea and that increase its iodine), and perhaps some egg whites if its not an autoimmune issue (although probably smart to avoid, seeing as it usually is autoimmune)
TLDR? - Answer: All 100% fresh grassfed meat cuts from the butcher, except minced meats should be all good. Even occasional freshwater fish, and poultry/ham.