I could actually be convinced relatively easily that god exists. All I would need is testable phenomenon where the existence of god is the most logical explanation. This acceptance would, of course, be tentative and further testing would be required.
For example, if all innocent people who were jailed would gain the ability to walk through prison bars, I would be convinced. Not that I claim to know who is innocent and who is guilty, but I imagine if maybe one to ten percent of the inmates couldn't be jailed due to supernatural abilities and claimed it was because of their innocence, i would believe them.
Or, if all the trees of the world (regardless of species) started producing fruit that had all the nutrients required to support life thus curing world hunger, I would be convinced.
That's not to say that I would be convinced that any specific god exists, or that my conviction wouldn't be quickly shattered if a natural explanation were to be found. Though, I think it would be hard to find a natural explanation for people being able to walk through metal bars ... which is probably why it hasn't happened yet.