Update 12-22-17
Hey guys, it's been brought to my attention that iodine may be something we want to consider adding to our sway as a prenatal-type vitamin. It ~probably~ doesn't sway either way (and if it does, it likely sways blue) but it is important for fetal brain development. I'm sure you've heard the expression "cretin"; this refers to a type of mental retardation caused by a lack of iodine during pregnancy.
Most people get ample iodine via iodized salt and prepackaged foods, and blue swayers might feel like, gee whiz, I'm eating a ton of salt already, do I really need this? But sea salt is actually low in iodine, eliminating dairy eliminates another source of iodine, soy iso and some other supps may block iodine from working as efficiently, processed foods don't usually use iodized salt, and on the High-Everything diet, you are prob. eating less processed foods anyway. So it's conceivable that you might end up eating less iodine than you would otherwise.
Please note, too much iodine is not good either and thyroid patients should NEVER take supplemental iodine without checking with your doctor first as it may not be appropriate for you to take. Iodine Intake and Effect on Thyroid Problems so doublecheck with your doctor to see if he/she feels an iodine supp is necessary.
If you do choose to take an iodine supp, the RDA during pregnancy/breastfeeding is 220-290 mcg a day, so no need to take any more than that (you may want to take LESS because you're already likely getting some via diet anyway.) Please avoid KELP supplements because they can be contaminated with arsenic and other grossness that would only sway pink (please obviously do not take arsenic for swaying pink!!!!) and be really bad for you anyway!
More info - Iodine | Linus Pauling Institute | Oregon State University
