I have always had the idea that God existed since I was a small child but never had what could be called "evidence" until I really started reading Intelligent Design arguments and that set a light off in my head/heart, so to speak. Intelligent Design is a completely rational, logical, scientific based argument for the existence of our Creator that is not dependent upon any kind of Scripture.
The only absolutely irrefutable evidence for God would be to see Him personally but that is not something we can do in this universe. It really comes down to a preponderance of the circumstantial evidence available to us, IMO. Some call that faith or belief and it depends on how much "evidence" an individual looks at or uses for the surety of such faith. For me, the structure of universe, the structure of our bodies and all living things, DNA, the structure of cells, microbiology, biology and chemistry all point towards a Creator. I always viewed the argument that there is an irreducible complexity to these things that random, purposeless, chance cannot account for. Hence the only rational, logical conclusion is God exists and He is the Creator. (this is a very abbreviated argument here)
Some very good, short, easily readable books on this is Darwins Black Box by Michael Behe and Signs of Itelligence by Behe and James Kushiner. They are scientists who put forth a non theological, scientific argument for the existence of a Creator and they make it easy for the average reader to understand the science behind it. For me, they were, among some other books too, absolutely profound in cementing my belief in the existence of God.