[I realise this post is long. If you want to read one paragraph only, choose the one with the bolded sentence]
What would convince me of the God of Christianity in particular? Evidence that I've been hallucinating every time I've had anyone describe how the Christian God is supposed to be.
Allow me to elaborate. While there's many, many things that don't line up with the Christian God hypothesis, I will use the argument from evil for the purposes of this post.
We are to assume an omnipotent entity which is supposed to be benevolent. There is no possible way that such a thing is even barely compatible with the universe we observe. For one thing, people die, all the time, for the stupidest of reasons, after living lives that may or may not be filled with suffering. Even if you buy the "free will" argument* that God cannot make all people be nice, the universe is simply not optimised for being good by any reasonable standard, for reasons that have fuck-all to do with human behaviour. Most of it is empty of intelligent life, for starters. If you could make the universe be any way you like, would you only put people in one particular place, unable to see almost all of it, guaranteed to die after a relatively minuscule amount of time? Why? Put us in every planet. Make us immortal, or able to live for as long as we desire until we get tired of it. Get rid of lightspeed limit, so we can reasonably see a bigger portion of the world. At the very least, don't add flesh-eating bacteria to the world, because what the fuck does that accomplish? I'm not saying I know the formula for maximising the goodness of human life, but sure as hell it doesn't look like this.
So between the choice that the universe is completely different to what I observe in basically every major way, and that I've simply managed to misunderstand every time I heard that God is omnipotent and he loves us, the latter is the least improbable one. Any universe where the God of Christianity exists as I think he's supposed to requires hypotheses on the Cartesian Demon level.
Any god in general is trickier question, not in the least because it's not a well-defined term. For the purposes of this, let's just say that a god is a being (with a mind, so as not to argue whether "the universe itself" counts as a being) that can perform straight-up miracles. Change the stars in the sky, manipulate time at will, that sorta thing. Let's ignore how they do this, so they could be an alien race that exploits technology we don't have or a Matrix Lord or a supernatural transcendent spirit that created the universe.
In that case, the most likely evidence that'd convince me would be them regularly using their miraculous power to accomplish their goals. If you change the midday sky to green with red spots once, then it's far more likely that I'm on drugs than that you are a god. If you do it every day to send messages to a friend in Mars, that's more interesting. If you teleport me to the moon for a few seconds to impress me, I don't know if you just have a very convincing holographic projector. If you can move the entire civilization across the galaxy because the sun is about to blow up, you're probably a god.
There are probably other things that would convince me, but the bottom line is that godhood should not be an evasive, immaterial thing that might as well not be there. A sentient being with god-level powers should have obviously visible consequences. If not, any clever excuse as to why we never notice their displays of godliness is by far more likely to be humans generating rationalizations than actual gods.
It's not impossible that there is a very good reason why some being that can reassemble the world at will, doesn't do so. Finding such a reason would greatly relax the standards of evidence, but would also mean by its very nature I'm unlikely to get that evidence. I don't know how an argument that proves that I shouldn't expect to see any displays of godly power (or see them but don't realise it), and yet allows that I will see evidence, would be like. Perhaps a personal meeting with a god-creature who wants me in particular to believe it is there but doesn't want anyone else to?
*I don't, but a universe where I realise my objections against it are invalid is the least unlikely hypothetical in this scenario.