This question gets asked a lot. The simple answer is of course evidence. Good evidence. Any good evidence would provide valid reasons to believe something is real. After all, this is why we accept that things like elephants, electrons, and traffic laws are real: we are all familiar with the evidence for those things (even if proving electrons would involve building devices that might be beyond the understanding of some).
Possible Convincing Evidence
I have in the past tried to consider serious evidence that would convince me. After all, if one takes a position, one should be prepared to consider what evidence of being wrong would look like so that we can always remain open to changing our minds in the presence of convincing facts.
One common answer that I used to give was "Rearrange the stars in the sky to spell something in a human language." It honestly doesn't matter what the message is. In order to make the stars appear to move, some of those stars would have to be moved faster than the speed of light tens or even hundreds of millions of years ago in order to appear to move right now. It would take a large number of astronomers a long time to verify the new positions of the stars to make sure it wasn't just some kind of optical illusion, but the movement of the stars could be verified through empirical means. Accomplishing this would require incredible power and planning.
There are other answers in the same vein, but I came to realize a serious problem with these answers. How could you distinguish between someone really powerful and all-powerful? The trick of moving the stars could theoretically be accomplished by a trickster alien with sufficiently advanced technology. While highly unlikely, we can't actually rule that explanation out. All of the other answers I used to give fall in this same category.
More importantly, we are all mixing up two proofs: proof of the existence of a being named god, and proof that said being is omnipotent, omniscient, and/or omnibenevolent. If a random homeless person approached me on the street and declared that they are god, that would certainly count as evidence of a god, just not the god any religion I know of claims to worship, because that random finite being is unlikely to have any of the characteristics we associate with the characteristics of any of the gods of actual religions.
Don't laugh at the above "evidence for god." While this might sound absurd, many people throughout history have accepted exactly that kind of evidence for gods, including most cults, the followers of Sun Myung Moon, the followers of that pedophile cult in Waco, Texas, and many other groups throughout history. If you want evidence for gods, there is lots of evidence for lots of gods, but not evidence that any of those claimed gods in history were any different from any other human being.
So what about proving omnipotence?
The problem is that I am a finite being in a finite universe, and that places strong limits on the sorts of evidence that I am able to empirically verify for myself. For example, if I were to meet a person who called themselves God, I could ask "Please create an infinite number of unique universes by twitching your left ear." While this would probably prove that said person has infinite power, there is simply no way that I could verify the existence of one more universe, much less and infinite number of universes, and I certainly could not verify that all of them were unique. The evidence would prove that a being has infinite power, but I don't see how I nor any other finite being could verify the evidence. As has been pointed out by others (I forget who originally said it, sorry), if an omnipotent, omniscient being existed, then he/she/it would know exactly what evidence would be needed to convince me that he/she/it existed, so it ultimately doesn't matter if I understand what evidence could prove such a thing.
But Does That Answer the Question as Posed By Theists?
In my experience, theists mean one or two things when they ask this question.
If You Don't Come Up with an Answer, That Means Christians Are Right
This is just the same, tired old "I have an answer and you don't, therefore my answer is correct" argument that often comes from theists. A large number of theist arguments reduce to this, and I have already covered this type of argument in another post, so I won't waste anyone's time covering it again now. No one owes you an answer to any of your questions. If you want to insist that certain things are true, you have to prove those things. Anyone's inability to answer one of your questions does not count as supporting evidence for any of your truth claims.
What Purely Logical Argument Would Convince You?
This argument is more reasonable and less tiresome, but ultimately no less bizarre than the above. So many classic theological arguments are pure logical constructs that operate without a shred of evidence. The "evidences" offered by non-theologians are generally very poor evidence (in the same vein as a random crazy person declaring themselves to be God), and these evidences are almost never mentioned by serious academics who study theology.
Those purely logical, evidence-free arguments are not just riddled with logical fallacies, it is bizarre that anyone would even attempt to offer pure logic as the evidence for the existence of a thing. Since theists are emotionally attached to the conclusion, in order to talk about why this is weird, we're going to have to trot out our old friend reductio ad absurdum and apply the same logic to a different conclusion so that theists can consider the logic without their feelings about the conclusion getting in the way.
Let us imagine that you met an adult who is reasonable, logical, but skeptical about the existence of elephants. Could you convince them that elephants exist using only logic and without citing evidence? I think not.
You can try the same mental exercise with other things. One could argue that the existence of a large number of motor vehicles would make traffic laws a tremendous improvement over not having traffic laws, but that doesn't mean traffic laws necessarily exist. For example, certain towns in Europe have removed traffic signs and traffic lights from certain intersections, and paradoxically, the accident rate went down. It's not hard at all to imagine a scenario in which someone figured out how to do that on a much larger scale and have safe roads without traffic laws. Or, we could imagine a scenario in which some group of people decided that traffic laws are an affront to their freedom and simply decided to accept a very high traffic fatality rate, similar to what Republicans did in response to the SARS-COV-2 pandemic. You could certainly try to argue that traffic laws are necessary, but you would not be successful.
I honestly do not think it is possible to prove the existence of anything using only logic and no evidence.
Once we allow evidence into the picture, it is trivial to convince people of the existence of elephants, electrons, traffic laws, the city of Cleveland, etc. Take evidence out of the picture, and it is likely impossible to prove the existence of anything.
What About Evidence for God?
There are plenty of people offering evidence for their god. Similarly, there is lots of evidence for alien visitation, just not any good evidence. There is plenty of evidence for the existence of faeries, just not any good evidence.
There is a similar problem with evidence for gods, but because theists are emotionally attached to the belief that their gods exist, it can be very difficult to explain to theists why their evidence is bad due to the nature of confirmation bias.
We are supposed to evaluate each argument and piece of evidence separately on its own merits, but we can accept bad arguments or evidence, or reject good argument or evidence based on whether or not it supports conclusions that we already accept. I cannot possibly overstate how incredibly insidious this cognitive bias is. I know what it is, and I know why it's wrong, and yet I frequently catch myself doing it anyway. With theists, it's even worse because there are so many strong emotions tied up with the acceptance of certain conclusions.
Theists will frequently recite litanies of evidence to other believers in order to reinforce both the conclusion and to make the supporting evidence sound more reasonable, but there is a reason that in apologetic debates between theists and atheists, theologians almost never discuss evidence. In order to have any evidence at all, theists have to lower their standards of evidence to such a shocking degree that all other religions become true at the same time theirs does, but if they raise their standards of evidence enough to eliminate the other religions, they also eliminate their own supporting evidence.
Do you notice improbable things? So do the practitioners of other religions. Do you have special feelings that let you know your beliefs are true and that the beliefs of other religions are false? So do the practitioners of other religions. Do you have "conversations" with the god or gods of your religion by creatively interpreting the meaning behind various events in your life? So do the people from other religions. Is every one of your prayers answered and the answer is always "yes," "no," or "wait/maybe"? The practitioners of other religions receive the exact same "responses" to their prayers by creatively interpreting the events that occur around them. Does your religion have "fulfilled" "prophecies"? So do many other religions. Once you understand what is wrong with both the prophecies and fulfillment of said prophecies in other religions, you will have the mental tools to start seeing the problems with the same thing in your own religion. Are there things in the universe that you cannot explain by any other way than your religion? The same is said by the practitioners of many other religions.
I get that there is a lot of evidence for your religion. There is also a lot of evidence for fairies, Bigfoot, alien visitation, vampires, werewolves, and all the gods of all the other religions. The problem is that there is no good evidence for any of those things, and that is why people reject claims of their existence. You may not reject claims about the existence of the gods of your religion, but you certainly reject the claims of the existence of the thousands of gods of the thousands of religions that are or have ever been practiced by man. It can be very tempting to embrace bad evidence because it supports conclusions that you like, but this is simply not a good way to establish truth claims.
The Answer to the Question is Evidence
As mentioned before, it would take relatively little to convince me that someone claiming to be god exists. It would take substantially more to convince me that someone has supernatural abilities. At the end of the day, all it takes is evidence, but it needs to be good evidence. Asking "How else do you explain me passing that math test?" does not count as valid evidence.