What does Isaiah 10:20 mean when it says a remnant of Israel will return?
It is very important when attempting to follow the prophecies, especially those of Isaiah, Jeremiah, Ezekiel and Daniel, to understand that the 10 northern tribes that formed the nation or Kingdom of Israel are called at various times 'Jacob', 'Israel' or 'Ephraim' but they’re never called 'Jews'. The nation or Kingdom of Judah is called at various times 'Judah', 'Jacob', 'Jews' and 'Israel'. You can get a better understanding of these terms by reading the books of Samuel, Kings and Chronicles in order to distinguish these two peoples in the prophecies.
The Bible verses in question are below.
A time is coming when the survivors (remnant) from Israel and Judah (the NIV, NASB and other translations have 'house of Jacob') will completely depend on the holy Lord of Israel, instead of the nation that defeated them. 21-22 There were as many people as there are grains of sand along the seashore, but only a few will survive to come back to Israel’s mighty God. This is because he has threatened to destroy their nation, just as they deserve. (Isaiah 10)
Preceding this passage (Isaiah 10:5-19) is a narrative of God’s punishment of the Assyrian Empire that had destroyed the Israelites (the northern 10 tribes that split off from the united Kingdom after Solomon’s death). God explained (verses. 5-6) that He used the Assyrians to punish the nation of Israel but those people were filled with false pride and decided to take on the nation of Judah (the remaining tribes of Judah, Benjamin and most of Levi - verses 7-11) and so they too would be destroyed (verses 12-19).
Assyria's destruction partly occurred when King Sennacherib brazenly threatened, in 701 B.C., to destroy Jerusalem. God killed 185,000 men of the Assyrian army when he sent the angel of the Lord to protect the city (2Kings 18-19). The Babylonian empire, in its own rise to power, destroyed what was left of the Assyrians and became the new world power when Nineveh fell in 612 B.C. Babylon, however, a short time later would come against Judah and Jerusalem in a wave of attacks that eventually led to the destruction of the city and the captivity of those in the Kingdom of Judah.
Some of those who lived among the ten tribes (the nation of Israel) escaped the Assyrians (only a few and mostly those originally from the tribes of Judah, Levi and Benjamin who still had families among the Jews – the nation of Judah) and they escaped into the kingdom of Judah. This is the 'remnant of Israel' of whom Isaiah wrote in verse 20.