Did the Old Testament predict Gentile inclusion in God's people?

Yes, explicitly. At least eight Old Testament passages anticipate Gentile inclusion, and the New Testament authors quote or allude to six of them as fulfilled — not as a new development but as the realization of what the prophets announced.

Yes, repeatedly and specifically. The Old Testament doesn't hint at Gentile inclusion — it announces it plainly from the very first covenant. When God calls Abraham in Genesis 12, he says:

"In you all the families of the earth shall be blessed." — Genesis 12:3

Paul quotes this in Galatians 3:8 and calls it "the gospel preached in advance" — he even coined a new Greek word for it, proeuēngelisato (προευηγγελίσατο, G4283), to make the point that this Abrahamic promise and the gospel are the same announcement, just separated by centuries.

That's one witness. There are at least seven more. Isaiah 49:6 says the servant will be "a light to the nations, that my salvation may reach to the end of the earth" — Paul quotes this at Pisidian Antioch as his direct warrant for preaching to Gentiles (Acts 13:47). Isaiah 56:7 says God's house will be "a house of prayer for all peoples" — Jesus quotes this when he cleanses the temple (Mark 11:17). Amos 9:11–12 says all nations will be called by God's name — James quotes this at the Jerusalem Council and says it's being fulfilled in the Gentile mission (Acts 15:16–17).

Ezekiel 47:22 goes further than most people realize. It says Gentile aliens who sojourn in Israel will receive an actual tribal inheritance alongside native Israelites — not spiritual blessing as a bonus, but covenant standing within Israel's own structure. And Isaiah 19:23–25 applies the phrase "my people" — the covenant-identity language God uses for Israel — to Egypt and Assyria.

Six of these eight passages are directly quoted or cited by New Testament authors as fulfilled. They weren't finding something hidden. They were reading what the prophets had openly said.

When Paul describes "the mystery" in Ephesians 3, he's not saying Gentile blessing was unexpected — the OT announced it constantly. The mystery was the mode: Jew and Gentile as fellow heirs in one body, through Christ, rather than Gentiles attaching to Israel.

For the full lexical analysis of these eight passages, see the study on dispensationalism, section "The Parenthesis: Eight Witnesses the OT Already Gave."