Hi Jeremy,
When we're hyperthyroid, fats move through our metabolism quickly and aren't absorbed by the GI tract very well. Consequently, we become deficient in oil soluble nutrients such as vitamins D, E, A and K and also CoQ10. CoQ10 is necessary for proper muscle function. Studies show that out of all medical disorders, the lowest CoQ10 levels are seen in hyperthyroidism. Levels are also low in people who take stain cholesterol lowering drugs, which also impede absorption. In hyperthyroidism, low CoQ10 levels contribute to muscle weakness. I don't know of CoQ10 having any effects on TSI, and I don't know of any commercial labs that run blood levels of CoQ10 (outside of research).
If you've been hyperthyroid in the past you may benefit from CoQ10 since deficiencies aren't automatically corrected when we become euthyroid. best, elaine