Quit drinking?

Thyroid problems are often due to a lack of iodine in the diet.
BUT radiation exposure does target the thyroid, and damage it wihout causing harm elsewhere.

Iodine is given in a large does to PROTECT the thyroid from radiation, in the even of accidental exposure.
I can't recall actual figures, but when there was the big nuclear explosion in Chernobyl, Russia, maybe 20 years ago, some neighboring countries dished out iodine doses to protect from the fallout, and some didnt.

The countries that DIDN'T found that their population developed a huge rise in thyroid cancer, while in the countries that DID give out iodine, there was zero incidence.

Its stocked in most developed countries to be used in the event of nuclear attack.

Are you sure that the iodine your mom takes is radioactive, and not just radio marked?
That is given prior to a scan to "show up", like a dye, helping detect problems that would not be seen in a plain scan.
These pills are usually given 24 hours before a scan, and the radiation is tiny, just enough to show up without causing any problems.