What is the name of this theological problem?

@David Stratton♦ answer is right. But also this problem can be part of the free will study.

This problem relatas to free-will because the ultimately, the answer to this problem is free will . So if you want to look further into that, i recommend you to study the problem of free will

I'd recommend this page for you: http://www.newadvent.org/cathen/06259a.htm

Problem of Hell seems closer to my question than Theodicy as I have seen it discussed.

When I was a teenager (I had been raised Lutheran), I confronted my parents with a summary of how I understood Christianity as had been presented to me by the church.

1. God created people with free will.
2. God punishes people for making the wrong choice.
3. The punishment lasts forever.

This summary seems to reflect God in a poor light. I was not so much condemning Christianity as I was asking for more information that would correct the thinking that brought me to such a negative view. Sadly my church and parents were unable to provide any clarity.

Over the years I came across ideas that greatly modified my original summary.

1. There is much that we do not understand, cannot understand, and worse, understand incorrectly.
2. The free will that God gives each of us will always result in us making the wrong choice. It is only the grace of God that brings any of us to Jesus.
3. Punishment in hell may only be judicially appropriate as opposed to real eternity (much depends on how aion is translated).
4. Churches, denominations, and individual preachers may not have as full a grasp of Biblical truth as they think.
5. The universe is a reflection of the God that created it. If it is not capricious, unreliable, or spiteful it may be that its creator is not either.
6. The law through Israel was supposed to show us how hopeless we were without Jesus. The kingdom of Jesus was supposed to show us fullness through the selflessness of love.
7. God established a system that works on both justice and love. Any emphasis on excessive punishment may reflect an inadequate understanding of the system God put in place.

Much of the reasoning that is able to put away the arguments of God's vindictiveness does not come from a deductive study of Biblical minutia. It does not come from an intuitive assessment of the experience of God. It comes from the work of the Holy Spirit in a heart that seeks truth.

1 Corinthians 2:14 But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned.

Those who would seek to understand have an open invitation from God.

James 1:5 If any of you lack wisdom, let him ask of God, that giveth to all men liberally, and upbraideth not; and it shall be given him.

In the Christian theology of Emanuel Swedenborg (1688-1772), which is one of my primary areas of expertise, the issues raised in your question fall under the general category of divine providence.

Swedenborg defines divine providence as "the form of government exercised by the Lord's divine love and wisdom" (Divine Providence #1).

Under that overall umbrella he includes his book on the subject the topics encompassed by theodicy, the problem of evil, and the problem of hell in the face of an eternal, omnipotent, and beneficent God.

For Swedenborg's approach to these problems, I highly recommend his book Divine Providence, originally published in Latin, Amsterdam, 1764. It is available in English translation, and in many other languages. For my recommendation of the best contemporary English translation, see my brief book notice and review here: Divine Providence, by Emanuel Swedenborg.

If nothing else, this book will help to round out your researches into this fascinating and highly debated topic.

In particular, Swedenborg distinguishes between "laws of providence," which define how God acts in relation to people whose lives are focused on what is good, and "laws of permission," which define how God acts in relation to people who are focused on what is evil.

For the good, according to Swedenborg, God provides every opportunity to continue moving forward spiritually so that after death they will live in the eternal joy of heaven.

For the evil, according to Swedenborg, God makes every effort to provide opportunities to reconsider, repent, and change course; but God will not violate the free will of the person by forcing a change of heart, mind, or destination.

In Swedenborg's theology, hell exists not because it is God's will that anyone should go there, nor because God desires to punish the wicked, but because humans who choose evil over good insist upon creating and living in a hell that reflects their own evil choices. The punishments of hell are not meted out by God, but are self-inflicted by the evil spirits themselves, and by other evil spirits in hell.

As in many other theological analyses of the problem of evil, in Swedenborg's theology it hinges on human free will, which in his view is a fundamental component of our humanity. I think you will find, though, that in Divine Providence Swedenborg offers a depth of analysis that is rare in traditional Christian and philosophical treatments of the subject.

Those with a more Christian Universalistic mindset find the problem of hell is not a problem. The subtitle of Beauchemin's book "Hope Beyond Hell" - "The righteous purposes of God's judgement" - suggests there is a purpose other than retribution.

You might also search for the unwritten sermons of George Macdonald - especially one called "Justice" - which is worth reading.