This question could be asked by two very different people. The skeptic seeks a kind of certainty based on unrealistic criteria for proving something is valid that not even science can live up to. The sincere questioner is truly exploring faith for themselves and they want to find more validation for their journey. You need to find out which kind of person you are dealing with before you respond with talking points.
By asking for certainty and proof, people are demanding stricter criteria for evidence than even scientists can demand… We cannot prove God exists, but we can offer good evidence for it.
The evidence that people are wanting in this question is empirical.
How we answer depends on the sincerity of the questioner. If they are skeptical, then they need to be convinced that the kind of evidence you can offer while not purely empirical, should be acceptable. If they are sincere, you can offer examples from Scripture and real life that give evidence that God is real.
By asking for proof, you are asking for more evidence than even scientists are expected to give. Scientists are reluctant to ever say a theory is “proven.” They put forward theories that they test by experiments which may verify their theory, but new facts may come along to disprove it later. People are demanding a “strong rationalism” that has been discredited as an argument against faith.
Faith is like getting married. You will never have proof that your future spouse will never cheat on you, but you can make a rational decision based on evidence you have about them. In faith as well, you can investigate all the evidence and make a list of pros and cons, but at some point you will need to take a leap of faith.
C.S. Lewis said, “I believe in God as I believe the sun has risen, not only because I see it, but because by it I see everything else.” So we cannot look directly into the sun by demanding irrefutable proofs for God. Instead, look at what the sun has shown you. We can find footprints of God’s existence all over nature: an ordered universe, human conscience, the big bang, the fine-tuning argument, the regularity of nature, etc.
The best evidence I have that God is real is the person of Jesus. All other religions are people searching after God. Only Christianity has God making himself known by becoming a person.
There is a lot of evidence for Jesus rising from the dead. The risen Jesus showed himself to over 500 people and there was an empty tomb, so it could not be claimed that his remains existed. Many of these eyewitnesses suffered and died as a result of proclaiming the resurrection, which is another validation of its truth.
Share how you came to believe that God is real and one way that he has changed you. Or, talk about how you see God at work in your community. I see God in my church community by inspiring us to launch a separate organization that provides health services to AIDS victims in our city.
Ask your friend if s/he would be willing to pray with you every day for a week, asking God to reveal himself to them.
Ask the questioner if they have looked at the accounts of Jesus in the Gospels to look at the actual evidence for themselves. (This is what started me to believe in Jesus!)