I have been trying to find more information on Hypothyroidism.
One interesting paper shows that there is a circadian rhythm to Thyroid hormones. Some practical implications of the graphs are
1) T4 is lower in the night, while TSH and T3 are higher in the night.
2) Sometime around 10:00 both TSH and T4 rise, while T3 dips.
3) The time when all 3 are steady for some time is afternoon 3:00 pm.
This means that the best time to get tested is around 3:00pm.
If you are supplementing T4, the best time would be when it is lower, ie night time. Also Mary Shomon reports that patients see the best results when they consume it just before sleeping. There is also a research paper published in 2007, where they saw significant difference between taking it in the night and taking it in the morning.
Remember the same strategy will not work for T3, because T3 is higher in the night. T3 is best consumed split up few times during the day, as it has a very low half life.
I came across a document explaining the various thyroid tests. It also has a table at the end summarizing the tests.