Yours is varying more than what I woudl think due to just circadian fluctuation woudl account for. Have you been tested for Graves, TPO antibodies and Thyroglobulin antibodies?
Any of those alone or in combination could cause you to have fluctuations like that.
Alternately, a pituitary tumor could cause weirdness like that too, though that is VERY unlikely.
As for the time of day that it is best to have TSH measured, supposedly, healthy people with a "normal" schedule have a surge of TSH starting about 9pm which peaks about 1am. In practice, though, we are all different. Some people have this peak in the afternoon and some in the morning, too.
Since TSH is released in several pulses once a day, it depends on when you get your pulse of TSH as to what you should do regarding meds, if all you are measuring is TSH.
I prefer not to rely on TSH, but to look at Free T3 and Free T4 to track my thyroid levels, just due to the daily fluctuations. If you are going by Free T3 and Free T4, I like to keep at least 8 hours between when the meds were taken and when the blood is drawn. If you have your blood drawn in the morning, take your meds after the blood is drawn. If your meds are drawn in the afternoon, schedule the draw as late as possible and take your meds as early in the morning as possible.
My advice may be contrary to what many doctors will recommend, but it is sound and is based on repeatable scientific evidence.
I am not a doctor, nor have I ever played one on TV...