How does our prayer play a part in predestination?
It is precisely by our prayer that [God] wants to remind us from whom we receive such benefits. For from whom do we receive but from him who it is fitting to ask… [The Church] prays that the incredulous might believe; hence, God converts them to the faith. She prays that believers persevere; hence, God grants them final perseverance. God foreknew that he would do all of this. And this is the predestination of the saints.
from St. Augustine Answers 101 Questions on Prayer by Fr. Ermatinger.
Predestination is the intersection between God’s omnipotence and His omniscience as they relate to creatures of free will within time. It is the solution to the question, “If whatever God wills, He does, how is it that some creatures ‘overcome’ His will for the salvation of all men?” The answer being that He wills the salvation of men generally by providing the means for the salvation of individuals, the is the sacrifice of the Son upon the Cross, yet chooses specific individuals to Mystical Marriage with the Son, which is accomplished through prayer by incorporation of the individual into the Church, participation in the sacraments, and living the life of grace wherein one is a temple of the Holy Spirit.
Prayer is integral to this. The ability to pray, the command to pray, and the content of the prayers come from God and are for all men. None of this is hidden but is the rain that falls upon the good and the bad alike. Yet the man who takes up the call to prayer did so because others have so prayed for him (the principal one who has prayed is Christ), others have so catechized him, and, as the man begins to pray himself, God begins to teach him in his own heart. It starts with God and ends with God. Prayer is the vehicle. Man, in humility and obedience, cooperates with that which God is willing and doing.
God wills that all pray but does not choose all to pray, nor all that pray are given final perseverance in their prayers. Those chosen are so because He omnisciently knows that if given the graces they will, so He omnipotently gives them the grace to do so. Those that would not, even if given the greatest gifts, He does not. This is a great mystery. Yet Christ prays for all. The Church and her saints pray for all as well.