It has been made abundantly clear to me that I should let meat rest after it has been cooked.
Once the meat is resting should I time the duration of the rest or measure the temperature of the cooling meat? And why?
The resting period is not generally done by an absolute temperature—you would have measured for (near) your target temperature before before taking the piece out of the oven or off the griddle or whatever.
Traditionally, it should be proportional to the size of the cut or roast.
If you do want to measure by temperature for more precise results, Kenji Alt of Serious Eats recommends based on practical experimentation, for steaks:
Ideally, no matter how well-done you've cooked your meat, you want to allow it to cool down until the very center has reached 120°F (49°C). At this stage, the muscle fibers have relaxed enough that you should have no problem with losing juices.
He also reviews some of the science, and various explanations for why the resting works.
Cook's Illustrated's Americas Test Kitchen performed an experiment with several roasted pork loins, cutting them with no resting, or resting of 10, 20, 30, and 40 minutes. They found that even a 10 minute rest cut down on juices expressed by 60% compared to the unrested roast. They found far less difference in the roasts that were rested for longer periods, although there was continued improvement in the juice retention. They summarize (original emphasis):
So what to do? Curb your enthusiasm and let your meat rest. But not for too long. The most dramatic decrease in moisture loss for these pork loins took place during the first 10 minutes of rest. Additional time helps—but not if that means your dinner will be cold. If you’re cooking a big roast, you can wait longer—about 30 minutes. Thin steaks should only wait 5 to 10 minutes before you dig in.
This shows that the traditional advise on resting times is actually fairly effective.
Note: Thanks to Jefromi for sharing this reference.