God does not reveal to us the method of his election and predestination. We do not know whether or not our prayers have an influence with God. There is a paradox here because God knows all things from all eternity. But on the other hand, James 5:16 says that the prayers of a righteous man will avail much with God. How can God from all eternity be influenced by those who would be praying for the salvation of others - through Jesus? Does God ordain our prayers as a means by which he saves them? It would seem so.
We are free in Christ and free to ask God to save others. We are free to preach the gospel and when we present the gospel to people we must understand that it has great power to save (Romans 1:16). When we proclaim gospel, we are proclaiming something very powerful and God undoubtedly uses it to bring people into the faith. But, again, he does not reveal to us how it all works. Our responsibility is to be faithful, preach the word, pray for the salvation of others, and trust that God in his great sovereignty uses our efforts, according to his will, to accomplish his purposes.