Virtually everyone agrees that the Holy books of religions (especially the Abrahamic religions) were written by ancient men. That is, there are no popular claims that God literally wrote these books. The claim instead is that God (Yahweh, Allah, Jehovah, ...) inspired men who then wrote the divine words.
A Credible Claim?
Most agree that men literally penned the Holy books. So what persuades us to go one step further and hold these books as the divine words of a perfect God?
One could show passages that illustrate knowledge beyond that of the ancient writers. For example, the specific prediction of an unlikely event that has since come true. Some argue that Genesis accomplishes this by 'predicting' the universe was created and that creation matches the Big Bang theory because it shows the universe had a beginning. But one would expect ancient men to presume everything is created. Cause & effect is core to human intuition – they simply wrote what was intuitively obvious. No prediction there. (Especially since the universe having a beginning does not mean it was created.)
An excellent example of advanced knowledge would be describing the Earth as one of a handful of planets orbiting the Sun. Or possibly explaining (with non-scientific language) that the lithosphere of our planet consists of moving tectonic plates which explain phenomena such as earthquakes and volcanoes. But no such knowledge is provided.
Alternatively, one might try to illustrate that a Holy book is perfect. Given human failings, it is almost impossible to produce something that is flawless but a perfect God certainly could accomplish this. A perfect Holy book would have no factual errors, no internal contradictions, no logical paradoxes, etc. Such a product would suggest the hand of a perfect entity. But no perfect Holy book exists; not even close.
There is a notable lack of evidence for the hand of a perfect entity but substantial evidence against. First and foremost we have clear errors. The Pentateuch (the first 5 books of the Bible) is common to Judaism, Islam and Christianity so it is a good book to consider. Immediately, we see Genesis describing the heavens as an ordinary bronze-age man might perceive it looking with his naked eyes. The sun and the moon are described as lights in the sky. The stars (and the visible planets by the way) are simply other lights. These men simply saw lights and had no cosmological knowledge. They did not know that our sun is a star, the moon is a rock that reflects light and that some of the lights in the sky are actually planets in our solar system reflecting sunlight. If the Bible had reflected something more than what the men with pens could see then that would be positive evidence of divinity.
Factual errors abound in stories such as Noah's flood which, among other things, describes an ark made of wood that we now know (by engineering and practical experience) would sink due to twisting motion of the water against wood.
Probably the most obvious evidence against the Pentateuch as the divine word of God are its many interpretations. If this work (and its subsequent Holy books such as the Bible and the Qur'an) is to be the word of a perfect, omniscient God then one would expect it to be so clear that it could not be misinterpreted. Yet the Bible alone has thousands of formal interpretations and individual believers –even within a specific formal interpretation (a denomination)– routinely disagree among themselves on fundamentals.
Beyond this, we have logical errors. The Bible defines an omniscient God (one that knows everything including the future). How is an omniscient entity surprised or disappointed? The Adam & Eve story, for example, depicts a disappointed God who casts his disobeying creatures out of Eden.
For vs. Against
The amount of evidence one can bring to illustrate flaws in books such as the Pentateuch (and beyond) is staggering. Biblical scholarship reveals the repurposing of ancient tales and the weaving of multiple inconsistent stories (see Noah's flood) into a single contradictory text. History refutes claims of ancient Jews in a 40 year exile, the stories of King David, etc. Science refutes the Genesis creation account, Noah's flood and countless other references. Logic refutes claims of omniscience and perfection in a God depicted as emotional, vain, petty and surprised by events of His making.
What evidence persuades anyone to hold these words by ancient men as the divine words of a perfect God?