How should we pray for unbelievers since God sovereignly elects people?
This problem is little different from the view that God simply knows who will be saved. Whether by omniscience or by decree, the future is settled—from one perspective. Thus, unless we want to deny God’s omniscience, we must live with this tension.
On the other hand, God has ordained the means by which people get saved: “How can they hear without a preacher?” Paul asks. It is always through human agency that God brings people into the kingdom. So the question is not how we should pray as much as whether we want to participate in God’s saving acts.
One other point: When it comes to healing, the scriptures say simply to pray for it—not to pray, “If it is your will...” I think that is the proper attitude toward unbelievers too—just pray for them, don’t ask the Lord to save them ‘if it is your will.’
There are many mysteries that we simply can’t fathom. But what we can understand is our duty before our Lord. We should act on that and trust him for the outcome.