Have you read about the gluten intolerance/celiac connection to autoimmune thyroid disorders?
Not saying this is YOUR problem, but it seems prevalent enough that if I had Graves disease, I would try 100% gluten free (tough to do, but often worth the trouble if it gets people off meds, on reduced meds, and/or feeling better--have a look at major celiac sites for instructions on how to go gluten free if interested) for 6 months to see if there were any changes. It takes 6 months most of the time to see any difference in various health conditions if they are caused by gluten.
Note that celiac disease many times exists with NO intestinal symptoms present...so...that's why I personally would choose to try a gluten free trial to see IF thing improved with this or lots of other autoimmune diseases that didn't go away after switching to McDougalling.
Don't know how to post a link, but I guess like this:
This should be enough to at least allow someone a launching point to see if they believe they would like to investigate this possiblity further for themselves with more research.