The thyroid gland produces a number of hormones that regulate metabolism, and as such both it and these hormones tend to show up in studies of aging and longevity. Variations in the operation of metabolism affect the pace of aging in individuals, and somewhere in the associated long chain of cause and effect can be found the thyroid and its activities.
One of the challenges inherent in building therapies based on data gathered about the operation of metabolism and variations between individuals is that it is hard to say whether what you are proposing to alter is a cause or a consequence. It is also presently a real challenge to discover all of the meaningful consequences of any particular metabolic alteration. So researchers can point to data and argue that longer-lived people tend to have characteristically similar levels of certain thyroid hormones, but you can't jump right from that to an assumption that trying to replicate this particular configuration of protein levels in other people will be beneficial.
This open access paper is a quick tour through published research, present consensus, and open questions regarding the thyroid and aging. A few of the more directly relevant portions are quoted below, but the whole thing is worth at least skimming for the examples of other associations with specific age-related conditions:
There has been long standing controversy about the thyroid function test results in the elderly. Serum thyroid-stimulating hormone (TSH), free thyroxine (T4), and free triiodothyronine (T3) concentrations change with aging. There is an increasing wealth of data suggesting that serum TSH levels increase with age, particularly after 70 years, but the data on free T4 is conflicting. This might reflect reduced end organ sensitivity, reduced turnover, and clearance, a genetic trait conferring a survival benefit or a combination of factors. In addition, no clear benefit is seen in treating a high TSH on a multitude of outcomes in the elderly. In fact, there is a possibility that treatment in the very elderly may lead to adverse outcomes. On the other hand, a low TSH has been associated with worse outcomes in the older age group.
The Leiden 85+ study showed that higher TSH concentrations and lower free thyroxine levels were associated with a survival benefit. In this study, participants with low levels of TSH at baseline had highest mortality rate, and participants with high TSH levels and low FT4 levels had the lowest mortality rate. The authors speculated that lower thyroid function may lead to lower metabolic rate which in turn could cause caloric restriction. Lower metabolic rate and caloric restriction have both been shown to be associated with improved survival in several animal studies.
There have been few recent studies exploring longevity with raised TSH and familial/genetic basis for this phenomenon. Data from the Leiden Longevity Study showed that when compared with their partners, the group of offspring of nonagenarian siblings showed a trend toward higher serum TSH levels in conjunction with lower free T4 levels and lower free T3 levels. In their extension to this study, they found that lower mortality in the parents of nonagenarian siblings was associated with higher serum TSH levels, lower free serum T4 levels, and lower free T3 levels in the nonagenarian siblings.
The comment on thyroid function leading to calorie restriction is interesting, because the relationship has been shown to go the other way: calorie restriction alters thyroid hormone levels in ways that appear similar to what is seen in longer-lived people. All these things are feedback loops in a connected system, of course, so it's quite possible to have cause and consequence in both directions.