I'd be very reluctant to insert iodine between your relative and her doctor. I wouldn't know how to predict any problems, and I think she's got all she can handle right now. It's sad to think that possibly the hyperthyroidism is due to being iodine insufficient, though.
Here are a few comments about users taking methimazole -- from Stop the Thyroid Madness
http://www.stopthethyroidmadness.com/2009/01/13/thyroid-tidbit-oprah/
http://www.medicinenet.com/methimazole/article.htm
DRUG CLASS AND MECHANISM: Methimazole is used to treat an overactive thyroid gland (hyperthyroidism). Grave's disease is the most common cause of hyperthyroidism. It is an autoimmune disease resulting from antibodies that attach to receptors on thyroid hormone-producing cells in the thyroid gland and trigger overproduction of thyroid hormone. An enzyme (peroxidase) produces thyroid hormones, i.e., thyroxine (T4) and triiodothyronine (T3), by combining iodine with a protein called thyroglobulin. Methimazole prevents iodine and peroxidase from their normal interactions with thyroglobulin to form T4 and T3. This action decreases thyroid hormone production. Methimazole also interferes with the conversion of T4 to T3. Since T3 is more potent than T4, this also reduces the activity of thyroid hormones. The FDA approved methimazole in March 1999.
GENERIC AVAILABLE: Yes
PRESCRIPTION: Yes
PREPARATIONS: Tablet: 5, 10, and 15 mg
STORAGE: Methimazole should be stored at room temperature, 15-30 C (59-86 F).
PRESCRIBED FOR: Methimazole is used for treating hyperthyroidism. It also is used for decreasing symptoms of hyperthyroidism in preparation for surgical removal of the thyroid gland or before inactivating the thyroid gland with radioactive iodine. Long-term use of methimazole may lead to a remission of the hyperthyroidism.
DOSING: The initial adult dose of methimazole is:
The daily dose is divided into three doses administered every 8 hours. The maintenance dose is 5-15 mg/day. The usual initial children's dose is 0.4 mg/kg given in 3 divided doses administered 8 hours apart, and the maintenance dose is half the initial dose.
DRUG INTERACTIONS: Warfarin (Coumadin) works by reducing the activity of vitamin K and therefore the formation of vitamin K dependent clotting factors. Methimazole may increase the activity of warfarin by further reducing the activity of vitamin K in the body.
PREGNANCY: There is evidence that methimazole may cause harm to the fetus during pregnancy.
NURSING MOTHERS: Methimazole is excreted in breast milk and may potentially cause harm to the infant.
SIDE EFFECTS: Methimazole is generally well-tolerated with side effects occurring in 3 out of every 100 patients. The most common side effects are related to the skin and include rash, itching, hives, abnormal hair loss, and skin pigmentation. Other common side effects are swelling, nausea, vomiting, heartburn, loss of taste, joint or muscle aches, numbness and headache.
Less common but more serious side effects include a decrease in white blood cells (agranulocytosis) and blood platelets (thrombocytopenia). Symptoms and signs of agranulocytosis include infections of the throat, the gastrointestinal tract, and skin with an overall feeling of illness and fever. Since platelets are important for the clotting of blood, thrombocytopenia may lead to problems with excessive bleeding. Hepatitis and death of liver cells (hepatic necrosis) have rarely been associated with methimazole.
Reference: FDA Prescribing Information