Although, this is a big subject, yet, let me try and make an attempt to answer it and if you are not satisfied with my response then please feel free to ask me a follow up question.
I think one of the challenges that people have when it comes to believing in the existence of God are of two kinds.
One, there are those who genuinely struggle (intellectually) or have reasons why they don't believe that God exists. In this case their main objection is, 'if there is such a person as a loving God, as the Bible claims that there is, then this God would have at least done away with the problem of evil. However, 'evil and suffering' is still around in our world. Therefore, they argue (philosophically) and conclude that this basically proves that there is no such thing as "God" particularly a 'loving one'.
Now, I am sure you would be aware that from the Christian point of view, philosophers have put forward positive reasons for why 'evil' still exists without negating the existence of God. Furthermore, Christian philosophers go on to argue that the Bible has very sound reasons for why God still permits evil in this world.
Anthony Flew is one example of someone who had intellectual reasons to believe that there is no God. With all the positive reasons that was offered to him by those who interacted with him, he later changed his view!
Secondly, there are those who finds it difficult to believe in the existence of God not because they have intellectual reasons for doing so but because they have an "emotional reason" or a 'dislike' or 'a hatred' or 'a barrier' in not believing in God. I have come across people who are angry with God perhaps because of some bad experience in their life such as the losing of someone very dear in their life.
For people with such emotional baggages, perhaps, an intellectual answer may not be of great help! Best way to deal with them is to provide support by means of 'listening' to them and to also pray for them as well as pray with them.