The thyroid gland carries out a number of important functions, responding to changing conditions by varying its production of thyroid hormones that alter the behavior of metabolism elsewhere in the body. The behavior of the thyroid changes with age, but in a sufficiently subtle and varying manner to make its role in aging a challenging thing to study. Nonetheless, there is at this point enough data to conclude that some forms of reduced thyroid function tend to associate with increased longevity in a number of species.

This also ties in with other lines of research. Calorie restriction, for example, reduces thyroid hormone levels in the course of extending life and improving health. A predisposition to low thyroid hormone levels appear to be inherited in long-lived families. And so forth.

Here is a short and very readable open access review paper that looks at thyroid function in the context of aging and longevity:

The thyroid gland and the process of aging; what is new?

The endocrine system and particular endocrine organs, including the thyroid, undergo important functional changes during aging. The prevalence of thyroid disorders increases with age and numerous morphological and physiological changes of the thyroid gland during the process of aging are well-known.

Intriguingly, decreased thyroid function, as well as thyrotropin (TSH) levels - progressively shifting to higher values with age - may contribute to the increased lifespan. [The] most striking findings concerning potential contribution of TSH and thyroid hormones to lifespan regulation, were obtained in the studies performed on centenarians (and almost centenarians). In 2009, Atzmon et al. published the results of studies on thyroid disease-free population of Ashkenazi Jews, characterized by exceptional longevity (centenarians). They have observed higher serum TSH level in these subjects as compared to the control group. [Moreover], the authors have observed an inverse correlation between FT4 and TSH levels in centenarians and [controls], and finally, they have distinctly concluded that increased serum TSH is associated with extreme longevity

The above-mentioned inverse correlation between FT4 and TSH in centenarians may suggest a potential role of decreased thyroid function in lifespan regulation, leading to remarkable longevity. Such a hypothesis seems to have been confirmed by the findings obtained in the Leiden Longevity Study, demonstrating the associations between low thyroid activity and exceptional familial longevity.

It should be stressed that reduced thyroid function with low levels of T4 is associated with extended longevity also in animals. For example, a very severe thyroid hypofunction with reduced core body temperature, as observed in Ames dwarf (df/df) and Snell mice [is] considered to substantially contribute to remarkable longevity in these rodents. [The] findings in animals are consistent with the results obtained in humans and may confirm a relevant role of thyroid hypofunction in lifespan extension.

Source:
http://www.fightaging.org/archives/2012/11/the-association-of-reduced-thyroid-function-with-longevity.php