What does Galatians 4:29 mean?

Paul is reaching the conclusion of his allegory, which contrasts living under the law of Moses to being justified by faith in Christ. He has shown this difference to be the same as that between living as a slave, or living as a free person. In this way, it is the difference between being born to Abraham's slave-wife Hagar (Genesis 16:1–3), as the result of human efforts, and being born to Abraham's free wife Sarah, as the result of the promise of God (Genesis 21:1–3).

Now Paul makes a further connection between the Ishmael/Isaac relationship and that between the Jewish religious leaders and the Christians of Paul's day. Ishmael, the son of the slave woman, persecuted Isaac, the promise-fulfilling son of the free woman. This persecution took the form of Ishmael, likely a teenager at the time, laughing at or mocking baby Isaac on the day of a feast in celebration of his being weaned (Genesis 21:8–9).

If we had been there, we might not have thought much about a teenage boy mocking his baby brother, but it was significant to Isaac's mother Sarah. That was the moment she demanded that Abraham cast out Hagar and Ishmael from their family to ensure that Ishmael did not share in Abraham's inheritance (Genesis 21:10).

For the purpose of his allegory, Paul is connecting Ishmael's mocking of Isaac with the Judaizers' persecution of the Christians. As Ishmael was born merely out of human effort, the Judaizers also taught that people could become acceptable to God through our own effort. Isaac's birth, though, was the result of God's work and the fulfillment of a promise. In the same way, Christians become God's children as the result of God's working through the Holy Spirit. It is not something we can make happen on our own.