The son promised to David is Jesus Christ, who will sit on David's throne, when it is restored in the era of his re-appearing on the earth.

Therefore being a prophet, and knowing that God had sworn with an oath to him, that of the fruit of his loins, according to the flesh, he would raise up Christ to sit on his throne; (Acts 2:30).

And, behold, thou shalt conceive in thy womb, and bring forth a son, and shalt call his name JESUS. He shall be great, and shall be called the Son of the Highest: and the Lord God shall give unto him the throne of his father David: And he shall reign over the house of Jacob for ever; and of his kingdom there shall be no end. (Luke 1:31-33).

Pilate asked him, Art thou the King of the Jews? And he answering said unto him, Thou sayest it. (Mark 15:2).

And Jesus said unto them, Verily I say unto you, That ye which have followed me, in the regeneration when the Son of man shall sit in the throne of his glory, ye also shall sit upon twelve thrones, judging the twelve tribes of Israel. (Matthew 19:28).

Of the increase of his government and peace there shall be no end, upon the throne of David, and upon his kingdom, to order it, and to establish it with judgment and with justice from henceforth even for ever. The zeal of the LORD of hosts will perform this. (Isaiah 9:7).

In those days, and at that time, will I cause the Branch of righteousness to grow up unto David; and he shall execute judgment and righteousness in the land. (Jeremiah 33:15).

Behold the man whose name is The BRANCH; and he shall grow up out of his place, and he shall build the temple of the LORD: Even he shall build the temple of the LORD; and he shall bear the glory, and shall sit and rule upon his throne; and he shall be a priest upon his throne: and the counsel of peace shall be between them both.
(Zechariah 6:12-13).