Sorry to hear your cat was diagnosed with hyperthyroidism. I've never heard of the iodine in the water thing to treat, nor would I recommend it. The Iodine treatment is radioactive iodine- so it goes to the thyroid and destroys the gland. Giving iodine in the water I don't imagine would do much as far as managing the disease.
1. Radioactive iodine- not going to be done here
2. Surgical removal of the thyroid glands- also probably not an option here
3. Methimazole (Tapazole) oral or topical treatment- this would be something to look into as it's fairly inexpensive and works well. I would make sure that the kidneys are functioning well before starting this therapy as if there is an underlying kidney disease it will make it much worse.
4. No treatment- this is essentially what is being done right now and not a bad option. Many people will have hyperthyroid cats that go untreated. They still do well, although you will expect a slightly decreased life span and you'll have to live with the hyperthyroid signs- hyperactive, weight loss, voracious appetite, unkempt hair coat.
Well that's hyperthyroidism and all possible treatment options. I'm not sure where they got the idea about iodine in the water but I've never heard of it. Can't hurt I suppose, but probably not doing much other than changing the color/ taste of the water.
Hope that helps, let me know if you have any other questions.