This is an issue that is near & dear to me personally, so I've done a lot of research, and tried lots of things. First, it helps to understand why the thyroid gets sluggish. Part of it IS just menopause, and for many of us it will correct itself if we can wait it out. An afternoon nap might be a good idea! Another factor may be the STRESS of the lives we lead -- our adrenal glands run on high for extended periods of time, get exhausted and call on the thyroid for backup, which eventually exhausts the thyroid. Another factor is environmental. We are bombarded with unnatural amounts of radioactive iodine from nuclear power plant emissions, nuclear accidents (TMI & Chernobyl) and nuclear test and bomb fallout. The trouble with taking Synthroid, or any other synthetic thyroid replacement drug, is that once you start, it is extremely difficult to stop. Long term use of these drugs has been tied to heart problems and loss of bone density. The dose your doctor is suggesting for you is very small, so I think you have a good chance of success with nourishing the thyroid.
So first, drink very little or no coffee. It puts stress on the adrenals. You can feel it, right? No white sugar or white flour. Use sea vegetables instead of salt. They are mineral rich, and contain good amounts of healthy iodine which protects your body from the radioactive stuff. Powdered kelp and Dulse flakes are my favorites. Dr. Ryan Drum, the herbalist who has done the most research into this issue, recommends Bladderwrack as the sea vegetable that is most effective in nourishing the thyroid & helping it to function properly. It has a pretty unpalatable taste, so I put the powdered herb into capsules. Dr. Drum recommends a combination of green, red & brown (kelp, Dulse and bladderwrack) sea vegetables as being the most effective way to nourish & protect your thyroid gland.
Don't eat raw cruciferous vegetables. This includes cabbage, brussel sprouts, broccoli, etc. Cook them. Don’t eat soy products unless they are fermented (miso, tamari). Use lemon balm and motherwort sparingly. These all have a tendency to depress thyroid function.
Drink deeply nourishing teas every day. Nettles, oatstraw and red clover is my favorite blend. Eat lots of cooked, dark green leafy vegetables -- chard, kale, collards, mustard greens. I like them sautéed until soft in olive oil with garlic & onions. When they are finished cooking I add some miso or tamari sauce -- Yum!
I have all of the herbs & sea vegetables that I've mentioned available on this website. Also, you might want to get a copy of Susun Weeds NEW Menopausal Years book, which has a chapter on thyroid issues in it. That's also available on my website.