Here's the scoop:
TSH varies daily by as much as 3 points. TSH is only released once a day, no continuously, like some hormones. In a normal healthy person with a normal schedule, TSH is released in a burst usually starting at 9:00 pm and ending shortly after the person falls asleep. People who are hypothyroid or who are night owls may release their TSH more continuously or at different times of the day.
SO, for a normal healthy person, if the goalk is to measure the TSH at its highest, it would have to be done in the middle of the night, or very early in the morning. IF the goal is to measure it at its lowest, then it is best to have it measured in the evening. Again, this ONLY applies to normal, healthy people with normal schedules. If your Circadian rhythm is off, the above will not apply.
T3 and T4 are another story. If you are taking meds, I don't advise you to take your meds withn 8 hours of having your blood drawn. The reason foir this is that after taking your meds T4 (and T3, too if you are taking a med with T3 in it) rises in your blood, unnaturally and peaks at about two hours, but remains unnaturally high for up to 6 hours before returning to where it should be.
Most doctors don't know this and therefore don't take it into account when they tell you to take your meds before getting your blood drawn.
My advice is to have your blood drawn as early as you can in the morning and then take your meds after getting your blood drawn. If you have to have i done later, take your meds as early as you can in the morning and then have your blood drawn as late as you can in the afternoon (take meds at 7am and blood drawn after 3pm for example). Your doctor's advice might be different as I explained above.
I am not a doctor, nor have I ever played one on TV...