Does The Curious Case of Benjamin Button have a moral, or attempt to teach its readers a lesson of sorts?

Fitzgerald's story is a meditation on the inexplicable nature of "life." I would argue that he's not so much presenting a moral as implying that life in its forward process (or progress) makes little sense. If it occurs in retrograde, backwards, it makes no less sense than in the normal forward way.
Button begins life as an old man then "youthens," (the term applied to Merlin in some versions of the Arthurian legends), finally becoming a baby and seemingly fading into oblivion: non-existence. The irony is that it's a mirror of normal life, because ordinary people begin helpless, become functional beings, and then end as helpless beings again in old age. It reminds one of the Sphynx's riddle solved by Oedipus: what goes on four legs in the morning, two legs at noon, and three in the evening? The answer is Man, and the riddle symbolizes the kind of cyclic journey of life, beginning with weakness, moving to a condition of strength, and returning to weakness again.
On one level, Fitzgerald is simply giving us an amusing conceit. The deeper meaning, however, is that this retrograde life span, simply because it is so bizarre, forces us to examine "real" life and to recognize how similar the forward process is to the backward one. In both cases, man sadly fades into nothingness at the end.