The mineral that is critical to the synthesis of thyroxine is

Thyroxine is the inactive form of hormones of the Thyroid Gland.
We commonly call Thyroxine: T4

The active form of the hormone is called triiodothyronine or T3. This is due to the mineral that is essential: Iodine

Without iodine, T4 cannot be converted to T3 and the hormone remains inactive. Iodine deficiency is common in parts of the world that have smaller amounts of iodized salt available for consumption, for example.

Therefore, the mineral that is critical to the synthesis of thyroxine (or T4) is iodine.