LOS ANGELES — On the morning of his first day of school, when he was 7, Abiy Ahmed heard his mother whispering into his ears.

“‘You’re unique, my son,’” he recalled her saying. “You will end up in the palace. So when you go to school, bear in mind that one day you’ll be someone which will serve the nation.”

With that preposterous prophesy for a boy growing up in a house without electricity in a tiny Ethiopian village, she kissed him on his head and sent him on his way.

Mr. Abiy, now 42, not only ended up in the prime minister’s palace. He has also become the most closely watched leader in Africa: a man who says he wants to change his country from the inside out — and fast.