Just how much do daily testosterone levels dip from peak to valley? This is an important question for many reasons, including the fact that almost every man will have his testosterone tested at some point. Sometimes men even get their own tests through independent labs as well.
So how important is the time-of-day in testing testosterone? It turns out that the answer to that question depends mostly on your age. Young males experience a substantial drop in peak daily testosterone, which occurs in the early morning, to the lowest 24-hour levels that occur in the evening hours. For example, one small, older study of young, healthy males found that testosterone peaked between 6-7 a.m. with levels of 670 ng/dl and then reached their nadir 13 hours later at an average of 464 ng/dl. 
This is a substantial 31% drop. A more recent study found a similar daily drop in average testosterone levels from 577 ng/dl to 420 ng/dl, a decrease of about 27%. 
What if you're a "little older" guy? One 2009 study addressed this and found that men in their 30's had average drops in testosterone of 20-25% from 8 a.m. to 4 p.m., which is about what one would expect based on the previous studies. (They were no measuring absolutely mins and maxs, but rather picking set times of day.) However, the same study looked at the above with 70 year olds and found that the difference was only 10%.  This means, effectively, that there is little difference in the typical senior's morning and evening testosterone levels.
What about free T and bioavailable T. Do they follow a similar pattern during the day? The answer is 'yes'.
These valleys and peaks in daily testosterone have to do with what is called the "cirdadian rhthym" of testosterone in males. Testosterone follow a cosine pattern - remember those trig classes? - and researchers have to handle this cosine pattern in their research as T levels to get higher in the early morning and lower in the evening.
Testosterone, in turn, is triggered by the pituitary hormone luteinizing hormone (LH). LH travels in the blood stream and acts on certain cells in the testes to trigger testosterone production. However, this whole process is gradually altered with aging, weight gain and other factors.
So the bottom line is that it does not matter as much if you are older when your testosterone is drawn, although it's a good idea to play it safe and get it drawn in the morning. However, it is very important to get your blood drawn in the morning, because one study found that about half of young men with hypogonadal testosterone reading in the afternoon actually had acceptable testosterone values in the morning.  Of course, do whatever your doctor tells you, but I have personally never heard of a doctor requesting a daily testosterone test in the afternoon for just this reason..
1) Journal of Clinical Endocrinology & Metabolism, Sep 1 1973, 37(3):366-371, "Integrated Concentration and Circadian Variation of Plasma Testosterone in Normal Men"
2) J Clin Pharmacol July 1, 2000 vol. 40 no. 7 731-738, "Modeling of circadian testosterone in healthy men and hypogonadal men"
3) J Clin Endocrinol Metab, 2009 March, 94(3):907 913, "The Effect of Diurnal Variation on Clinical Measurement of Serum Testosterone and Other Sex Hormone Levels in Men"