[For background see: http://lesswrong.com/lw/n8u/a_medical_mystery_thyroid_hormones_chronic/
I thought of a class of solutions, I went looking for possible evidence, someone's already proposed what looks like a perfect answer, and the problem is much bigger than I originally thought.]
[ Epistemic Status 1: Gather Underpants 2: ? 3: Profit! ]
(1) Some common mechanism(s) can interfere with the reception of the endocrine hormones by the cells on which they should act.
There would be a high genetic load on such a mechanism, so we should look for recent environmental change, immune defence, or incomplete adaptation to less recent environmental change for the causes. 
Seek, and you will find: Such a mechanism was proposed in 2003 in:
A metabolic basis for fibromyalgia and its related disorders: the possible role of resistance to thyroid hormone R. L. Garrison, P. C. Breeding
These authors may have seen the whole of the truth for all I know, it looks terribly plausible to me, but I don't understand any of the interesting words in their paper. Hyaluronic. Now there is an interesting word. I wonder what it means. Nevertheless, this is exactly the sort of thing we should be looking for. I would imagine that there might be more than one such mechanism.
Then we would expect to see something like the classical presentations of endocrine disorders without any evident disturbance of the endocrine hormone levels in the blood.
Consider for instance Hypothyroidism / Hypometabolism / Myxoedema, a form of general metabolic collapse disease with famously many symptoms which appear almost at random, famously difficult to diagnose.
Pick a symptom of Hypometabolism and suppose it your primary symptom: For instance T3 deprivation in cells reduces the ability of mitochondria to recycle ATP, resulting in complete, shattering exhaustion from the mildest exercise.
Take this to your doctor. If competent she will test you for hypothyroidism (and all other common causes of fatigue). Your test will show that your blood hormone levels are normal. At this point, you have a mysterious unexplained syndrome in which the primary symptom is chronic fatigue, but which overall shows similarities to hypothyroidism. You have Chronic Fatigue Syndrome.
Suppose that the symptom that bothers you most is widespread pain. Then you will eventually be diagnosed with Fibromyalgia.
Should you complain mostly about alternating constipation and diarrhoea, then you have Irritable Bowel Syndrome.
Hypothyroidism, being a general collapse of the metabolism, can present with about forty different symptoms.
We would expect to see a number of overlapping 'syndromes', all with different primary symptoms, but all with great overlap with one another, and with the ancient and no longer understood metabolic collapse syndrome associated with Hypothyroidism, once familiar to doctors but no more.
We should also see various other overlapping clusters of syndromes, associated with random tissue deprivation of different endocrine hormones.
We should see that these syndromes have exploded in prevalence since 1970, when diagnosis of endocrine disorder by clinical symptoms went out of fashion in favour of diagnosis by blood hormone level tests.
We should see low levels of abnormal thyroid blood tests in these populations of sufferers, because some diagnoses of classical hypothyroidism will have been missed. But on the assumption that most doctors are competent, these levels should be above the general population levels, but not nearly high enough to indicate that the symptoms are caused by thyroid disorders.
If one of the obstructing mechanisms is immune in nature, then we should see these various disorders occasionally appearing shortly after infections. Particular types of infections should be more likely to cause them than others.
I believe that that is exactly what we see. They are known as the 'somatoform' disorders, because they are thought to be all in the mind. By those who have never had one.
I have a feeling that the air of crankiness around Lyme Disease, and the belief that its chronic-fatigue-like symptoms get worse long after the known infective agent has gone, might be explained in this sort of way.
But I am tempted also to include other mysterious diseases without known causes and with symptoms plausibly explained by endocrine hormone abnormalities, such as Bipolar Disorder, Depression, and the 'Metabolic Syndrome', which may do exactly what it says on the tin.
In particular, it is known that the principal characteristic of Chronic Fatigue Syndrome is Mitochondrial Dysfunction  . I contend that this is principally caused by lack of the hormone T3 in cells, for reason or reasons currently unclear.
The TSH test in particular is suspect, since it appears to have been justified on the basis of a simplistic model of the thyroid hormone system which had very little explanatory power even at the time, and which is now known to be a hopeless oversimplification. Even allowing for this context, the sensitivity of the TSH test never seems to have been investigated.
It is well known in the alternative medicine community that thyroid hormone treatment can alleviate Fibromyalgia and Chronic Fatigue. Some put this down to the 'stimulant action' of the thyroid hormones, believing that a similar effect would be achieved with amphetamines. But this is known to be untrue. A 2001 trial by some brave Scottish GPs proved conclusively that thyroid hormones have perceived harmful effects on healthy people .
The fact that this has been taken as a refutation of the alternative medicine idea of treating Chronic Fatigue Syndrome with Desiccated Thyroid is most unfortunate.
We therefore see that (1) =>
(2) There is a huge, generalized, common disorder with many names, which is caused by inadequate thyroid hormone stimulation of peripheral tissue.
(3) There are further clusters of disorders corresponding to other hormones.
(4) These clusters themselves may overlap. Whatever interfering mechanisms there are may interfere with many hormones at the same time.
Following the suggestion of Garrison and Breeding, by analogy with the situation in diabetes, I call the disorder in (2) type II hypothyroidism. It is not to be confused with central hypothyroidism, which is detectable by blood hormone tests, although not by the TSH test. John Lowe called this disorder 'peripheral resistance to thyroid hormone'.
Since (2) would be such a good explanation of observed patterns of mysterious diseases, it becomes urgent to refute the hypothesis (1)
How to Refute the Central Hypothesis
We seek sufferers of type II hypothyroidism amongst the sufferers of Chronic Fatigue Syndrome and Fibromyalgia.
I choose Chronic Fatigue Syndrome because I have had it myself, and thyroid hormones have so far had an excellent effect on me, including raising my basal temperature to normal levels.
I choose Fibromyalgia because John Lowe dedicated his life to establishing that the symptoms of Fibromyalgia and Hypothyroidism were one and the same, and to apparently successfully treating sufferers of Fibromyalgia with thyroid hormones.
We filter out all those with abnormal blood hormone levels. They are classically hypothyroid and should be treated as such, although the possible presence of interfering mechanisms must be remembered, and the treatment should be by symptoms and not by hormone levels.
In our remaining population of CFS/FMS sufferers with normal lab values, I expect to find many people with the classical symptoms of hypothyroidism.
We could score them with the Billewicz test , the last word in clinical diagnosis. Although note that by design this test does not take account of the most obvious hypothyroid symptoms!
Or we could score them by what John Lowe considered the principal symptom of hypothyroidism, the ratio of measured basal metabolic rate to the metabolic rate predicted from such factors as weight, age, and sex.
I propose that we do both and I expect that:
(5) In the CFS/FMS population, there is a proportion of sufferers with abnormally low metabolic rate and abnormal hypothyroidism scores.
If (5) is not true, (1) is refuted. My beautiful if somewhat disturbing hypothesis refuted by an ugly fact, I shall shut up about it and start thinking about another way to explain the mystery.
If (5) is true, then we may wish to consider attempting to treat these conditions with desiccated thyroid, since that is what everyone who cares about these diseases has been telling us works since about 1940.
 Infectious causation of disease: an evolutionary perspective Gregory M. Cochran, Paul W. Ewald, and Kyle D. Cochran
 Is rheumatoid arthritis a consequence of natural selection for enhanced tuberculosis resistance? James L. Mobley
 A metabolic basis for fibromyalgia and its related disorders: the possible role of resistance to thyroid hormone R. L. Garrison, P. C. Breeding
 Chronic fatigue syndrome and mitochondrial dysfunction Sarah Myhill, Norman E. Booth, John McLaren-Howard
 Thyroxine treatment in patients with symptoms of hypothyroidism but thyroid function tests within the reference range: randomised double blind placebo controlled crossover trial M Anne Pollock, Alison Sturrock, Karen Marshall, Kate M Davidson, Christopher J G Kelly, Alex D McMahon, E Hamish McLaren
 Statistical Methods Applied to the Diagnosis of Hypothyroidism by W. Z. Billewicz, R. S. Chapman, J. Crooks, M. E. Day, J. Gossage, Sir Edward Wayne, and J. A. Young