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:
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):
This shows that the traditional advise on resting times is actually fairly effective.
Note: Thanks to Jefromi for sharing this reference.