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 cannot be consistent with the self-revelation of God in them (and Universalism must be seen to bite the dust as well).
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 faith to spark into life.
Really, God "showing up" is that enabling help without which no one could truly believe, but I also know that God, no matter what help he gives, isn't going to believe for us. Otherwise, all of his commands to us to believe would be nonsensical. In order to effectively trust in Christ, only a minimal amount and clarity of faith (i.e. less than mustard seed quantities) is required. That may not seem like much of a threshold, yet it remains a mountain to most people most of the time.
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. 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 by Israel's example and the fact that not everyone comes to faith since Jesus was lifted up on the cross. Nonetheless, thank God that the Holy Spirit is sent to bring us to that moment when everything comes together and Jesus is seen as supreme--the faith moment.