I do not believe that God secretly presses a button he's concealed within us, which when pressed, makes us people of persevering faith. As I understand it, that is precisely what Calvinism proposes. The problem with that is that if God did do that kind of thing for any person, he'd do it for all people. Scriptures are clear that is not the way things turn out, so Calvinism's view of efficacious grace cannot be consistent with the self-revelation of God in them.
God has made mankind with the capacity for faith, of that there can be little doubt, for people everywhere trust in things they cannot see. I think this general capacity is what separates mankind from angels, particularly in regard to redeemability. Mankind was made in innocence, really ignorance, and therefore was made for faith. Faith exists in that gap produced by unseens and unknowns, but Angels were made for knowledge and sight.
When angels rebelled they did so in knowledge and sight and are irredeemable as a result (see Hebrews 6:4a for the concept as it applies to mankind). If Romans 12:3 applies broadly to all humanity (as I've always taken it to mean) rather than just the church (as Calvinists in particular take it), then God has in fact dealt each person at least some measure of faith. Of course, true faith, faith that actually has an effect, requires that it be placed in the right object, namely, God and God alone. That means that God has to "show up" for that faith to spark into existence.
God "showing up" is that enablement without which no one could truly believe. But God, regardless of what help he gives us, isn't going to believe for us (which is what irresistible grace is tantamount to in my mind). All of his commands to us to believe would be nonsensical in that case. No, it is we who must trust in God, that is our God-enabled responsibility.
We are called to faith, it is the very currency of heaven. On their own, humans can only answer that call with something less than true faith in the actual God. However, when the Holy Spirit brings our focus on the person and authority of Christ into clarity, the moment is ripe for salvivic faith to be born. It is not guaranteed, as is attested to by Israel's example and the fact that not everyone comes to faith since Jesus was lifted up on the cross, but is only possible then and impossible otherwise.
Nonetheless, thank God that the Holy Spirit is sent to bring us to that moment--the faith moment, when everything comes together and Jesus is seen as Savior and Lord.