Generally speaking, flexibility training is enhanced when your body is warm; so morning is usually not a good time for most people. You could, however, do some very effective range of motion exercises that will assist you in your goal, in the long run.
For the actual stretching, dynamic warm-up moves will promote some flexibility. However, the deeper, longer held stretches are best done once your core temperature is elevated, usually later in the day and after some type of activity, like walking, running, etc…
The best time of the day to perform flexibility stretching is ANYTIME! Whatever is convenient for you. If it’s convenient, you will do it. The best way to be most effective is when the muscles are warm and receptive to change.
Having said that, you need to choose a time of the day that is convenient for your schedule to warm the muscles and stretch them. For example, right after a workout, or after peforming gentle motions of each muscle through its range of motion (dynamic stretching), or after a warm bath/hot tub.
Just like curling your hair….if you heat it first, then wrap it in a curler (or something similar), as it cools down it will have a greater chance of taking on that shape.
With muscles, if you warm them first, then slowly and gently stretch each one to it’s fullest ability, and hold each stretch for a minimum of 30 seconds (static stretching), then as it cools down it will be more likely to sustain its maximum flexibility.
Caution! Just like balancing a strength training workout (i.e. working the front half of the body as much as the back half – working the right side of the body as much as the left – working the top half of the body as much as the bottom half), you similarly need to balance your flexibility workout; stretching mucsles of the front and back half, right and left halfs, as well as top and bottom halfs.
Conventional wisdom says both before you workout or participate in a sport, AND after. Dynamic flexibility exercises before and static flexibility/stretching after. Of course you can also stretch (and should) on days that you are not actively working out or playing a sport. In those cases, I prefer a light static stretching (stretch and hold) in the mornings before getting my day started, and a longer static stretch in the evening before going to bed. Feels great!