It seems your friend has gotten good advice from her conventional doctors, presented in a poor way. Without knowing the naturopath in question, you can’t say if he is one of the ones that respect and use modern medicine in additional to other strategies or one of the many who rely solely on woo. It’s similar to chiropractic in that there is some evidence that chiropractic can help back pain and there are good chiropractors who use manipulation to alleviate musculoskeletal back pain. However, when they start saying that they can cure cancer by aligning the spine you are in trouble, and you can’t know how much of their spiel to believe. Certainly, taking a skeptical approach to a natural doctor cannot hurt. At the least, find out what his theory of disease is and if he starts talking about removing toxins and balancing humors, you know you are in trouble.
Not to give medical advice online, but if your friend has Graves disease, she will eventually be trading hyperthyroidism for hypothyroidism whether she wants to or not. That is the normal course of the disease. Eventually, the thyroid will become underactive. Like her doctors told her, she has 3 choices:
#1) Take medication, which works in 30-50% of people to control the symptoms until the hyperactive stage resolves (usually in about a year). If it works, then she gets watched periodically until the time, 10, 20 or 40 years from now when her thyroid stops working and she eventually will need supplements, unless something else kills her first.
#2) Take radioactive iodine to destroy the thyroid (and stay away from pregnant women and infants for a week) and then take supplements to give her the right amount of thyroid for the rest of her life. They can aim to destroy just enough thyroid so that the remaining amount works normally, but it is very hard to judge, and as noted above, her thyroid will eventually become underactive anyway so most doctors err on the side of overtreating.
#3) Surgically remove the thyroid and take the medication as above. Again, you can aim to remove just enough of the thyroid to have normal function, but you risk undertreating and having to retreat or overtreating and having to take supplements anyway and since the patient will eventually end up on supplements anyway, it makes sense to overtreat.
In short, your friend unfortunately likely has a sick thyroid and eventually she will need to be on supplements. It sounds like she is having a hard time dealing with this. As noted above, she has to think of it as an essential nutrient that she needs to live, vitamin T as it were. I think that once she accepts the fact that she will likely eventually need daily thyroid supplements for the rest of her life, she will feel better about her options. Also, she needs a doctor with a better bedside manner.