Intellectually: David Hume is highly regarded by many atheists by declaring that anything beyond natural law is impossible. Or to put it another way, it seems Hume was saying that he knew the limits of all knowledge. Yet today even the most egotistical person will admit that 'we' (humans) don't know .000000000001% of the how, why, what, where, who, when of existence, to borrow from Kipling. Even in Hume's day critics charged that he was using circular reasoning; and that he didn't reply to his critics.
Intellectually and experientially: Dr. Craig Keener grew up an agnostic. One day during his teens he met some rather heavy-handed Christians: "You're going to Hell unless...." He dispatched them with hard counter punches. By that time, he thought to himself, these Christians aren't even serious about God. If I knew God existed, I would give it my all. That night he claims that he had a supernatural experience, the presence of God. That jump-started his life-long quest.
Years later he would be an extremely prolific author. His study of the Book of Acts, 3 million words, has been called 'magisterial,' 'epic,' 'the very best study of Acts....' He also wrote a 600,000 page study of miracles. Many of the cases have full medical documentation. He features over 300 cases of blindness alone.
One of the most famous cases involved a teenage girl with a most severe case of multiple sclerosis. Over the years until she was a young adult, her physically condition deteriorated. At the Mayo Clinic, one physician said she was among the worse cases of MS he had ever seen (the names of the MDs and other witnesses are included). When she was given six months to live (she was sent home to her family) her body was twisted, her feet pointed downwards, similarly with her hands. Her intestines were shutting down, she lost all control of her basic bodily functions. She was nearly blind. Yet through all this she was a Christian.
At this point she had lost all hope, that God didn't hear her prayers. Her story was featured in a radio program. Hundreds of listeners began praying for Barbara. Her cousin was reading some of the letters from these listeners. And then Barbara gestured to her cousin that she wanted to say something. She said that she knew God was going to heal her, and to call her parents. Her mother ran in to find Barbara had ripped out all the tubes and was standing on legs that had been atrophied. Her mom bent down and felt her legs, "You have muscles!"
Her dad began to waltz with her around the room. At a prayer service at her church that night, the minister asked if anyone had anything they would like to add. Barbara began walking down the center aisle, and you can imagine the shock and incredulity. Everyone began singing, 'Amazing Grace.' This, of course, is the famous hymn written by a life-long atheist, who in his own words was the most wretched of men, until he was saved by God.
Another scientific study (not as hard to read as 600,000 pages) is from Dr. Candy Brown. She and a medical team went to Mozambique, East Africa to test Heidi Baker, a missionary there for decades, and the site of many alleged miracles, including miracles of healing.
Over two dozen patients were brought and tested, either legally blind or deaf (because these cases can be pre-tested easily); and then Heidi Baker would pray and hold the patient for a few minutes. Immediately afterwards, the patient would be tested again. The majority of the patients experienced incredible healings. For example, 'Ruth' was an elderly woman with vision tested 300/20 and so deaf that a jet engine couldn't faze her -- plainly she couldn't hear a normal conversation. A few minutes later her vision was 80/20 and she could carry on a normal conversation. Dr. Brown explains the misconception that with so few test subjects the study wouldn't be an 'acid test.' However, with the fewer number, the results are actually more astounding. Her husband also has founded the Global Medical Group to scientifically test the effects of prayer on illness.
A study of over 1,000 American MDs shows that over 70% believed they had been involved in cases that could not be explained by naturalistic means, that they were open to the possibility these could be classified as miracles.
If one is of an intellectual frame of mine, I'd suggest Dr. Craig Evans' 'Fabricating Jesus - How Modern Scholars Distort the Gospels.' Evans' 'How God Became Jesus' also is an excellent refutation of Dr. Bart Ehrman's 'How Jesus Became God.' Drs. Gary Habermas (resurrection expert), Edwin Yamauchi (a 'scholar's scholar'), Richard Bauckham (one of the tops in 'early high Christology club' an informal 'club' with a focus that Jesus was worshipped as God from very early after the resurrection) are one of several scholars I would recommend that use the strictest methodology. Dr. John A. Bloom, physicist, has written a fine study of Ezekiel's prophecy of Tyre, which skeptics claim is false, along with the rest of the Old Testament prophecies. Bloom also has written a top 80 page paper on the existence of God.
On a concluding note, the message of God as revealed in the Bible is of love. In 1 Corinthians 12 + the apostle Paul encourages believers to seek the highest Gifts of the Holy Spirit. These are supernatural gifts (sorry, Mr. Hume). While the list is varied, it may not be complete. For example, in times of trouble, sometimes a Christian being physically attacked has recounted the attacker being immobilized by an unseen force, such as the case of a Soviet-era Christian hunter who regularly beat up Christians. After experiencing these things, he became a Christian.
The highest Gift, though, is love. "Faith, Hope, Love ... the highest is Love." And from this we can understand how normal humans can be capable of incredible love. The 'fruits' of the spirit are developed by the normal walk through life of the Christian who learns about God, and tries to apply the lessons. But with the Gifts, God helps one supernaturally.
Paul also writes that 'God is love.' Despite all the skeptics, you can prove to yourself if this is true or not. And that is a brief answer on how an atheist can come to belief in God.
