What is the meaning of the phrase “The morning constitutional”?

It literally means "something that is good for your constitution", usually a walk, but it's also a common euphemism for the first visit to the washroom, particularly in areas where heading out to use the facilities is not a very distant memory. It was much more common in my youth (and it's been a while since I was a youth) among older people, and it seemed to carry a bit of feigned poshness among the working class (who didn't need to go for a purposeless walk to get their exercise).

The New Oxford American Dictionary has for constitutional:

noun (dated): a walk, typically one taken regularly to maintain or restore good health.

Regarding etymology, constitution means “a person's physical state with regard to vitality, health, and strength”, so the constitutional comes from its supposed benefits to the health. (Think of it as a “walk to improve one's constitution”, if you will.)

It also frequently refers to a morning bowel movement. Its a euphemism for a bowel movement. (From a time when using the bathroom meant going for a walk to the outhouse--the original meaning was still "going for a walk", but this idea was used in the euphemism for going to the bathroom.)

Maybe this is an American English versus British English or regional thing. I've never heard the phrase used to mean a walk. I've only heard it used as a euphemism for a visit to the bathroom.