The name
"Who say to Cyrus: Thou art my shepherd, and thou shalt perform all my pleasure. Who say to Jerusalem: Thou shalt be built: and to the temple: Thy foundations shall be laid."
Isaias (Isaiah) 44:28 · around 700 BC
Isaiah doesn't say "a king will come." He gives the man's name. Then he keeps going:
"Thus saith the Lord to my anointed Cyrus, whose right hand I have taken hold of, to subdue nations before his face, and to turn the backs of kings, and to open the doors before him, and the gates shall not be shut."
Isaias (Isaiah) 45:1
The problem
When Isaiah wrote this, Jerusalem was standing. The Temple was standing. Nobody needed to rebuild anything.
It would be another hundred years before Babylon destroyed them, in 586 BC. Isaiah was describing the rebuilding of a city that had not yet fallen, by a king who had not yet been born, from an empire that did not yet rule the world.
What happened
Cyrus became king of Persia and swept through kingdom after kingdom. In 539 BC he took Babylon itself, and the greatest city in the world fell to him without a battle.
The next year, he signed this:
"Thus saith Cyrus, king of the Persians: The Lord, the God of heaven, hath given to me all the kingdoms of the earth, and he hath charged me to build him a house in Jerusalem, which is in Judea."
1 Esdras (Ezra) 1:2 · 538 BC
He read it himself
The Jewish historian Josephus records that Cyrus read Isaiah's prophecy about himself. Moved by it, he set out to do what was written (Antiquities 11.1.2).
Cyrus's own record of freeing captive peoples and sending them home survives too. It's carved on a clay barrel called the Cyrus Cylinder, now in the British Museum.
Why this is a big deal
Anyone can predict that a city will be rebuilt someday. Isaiah gave the name of the man who would order it, before the city was even destroyed.
God called a pagan king "my anointed," and the king answered to his name.