I am going to first walk you through Galatians 3 starting in verse 6.
Paul first makes the point that Abraham believed and because of his faith he was credited with righteousness. He then immediately says (verse 7), “Therefore be sure that it is those who are of faith who are sons of Abraham.” He continues to emphasize that it is through faith, not through the Jewish Law, that one shares in the blessing of righteousness that was given to Abraham and then says explicitly in verse 14 that “we would receive the promise of the Spirit through faith.”
In order to further clarify this point he said that the promise (that all the nations would be blessed in him – which Paul defines as the Gospel in 3:8) was given to Abraham 430 years before the law had been given (vs 17). Therefore, inheritance of the blessing (salvation/justification/the Spirit) was not to be based on the law: Having the law or following certain aspects of it. (Aka being a Jew and having the law was not a requirement or the way in which one could receive this blessing). Rather, just as this promise of blessing was given to Abraham because of His faith (which James 2 clarifies is a faith that affects one’s actions and entire lives and is not just an intellectual belief of his deity like the demons have), the promise of blessing would also be given to all those who would believe (Gal 3:22).
He says that if we have faith like Abraham, we are sons of God, we are in Christ and true sons of Abraham –heirs according to promise. (Gal 3:29)
Therefore, by using the language of “child of promise” in Romans 9 Paul is bringing us back to the promise He gave Abraham – Which was before Israel had been formed, before the law had been given, and before circumcision was commanded. It is according to that promise that the true children of God have been birthed. And therefore Paul calls these true children of God, those that have faith like Abraham, children of promise in Romans 9.