I recently had a conversation with a self-centered hedonistic American teenager who was not interested in hearing proofs that the Torah is the word of God and said he could serve God any way he wanted. He asked me why he should be bothered practicing Judaism with its restrictions which don't speak to him as having any religious value. I would recommendations for books to read (either myself or him), podcasts to listen to or other media to consume which offers a non-intellectual based, teenage oriented response to why someone should be an orthodox Jew.
One name that comes to mind is Zecharia Wallerstein. There are a tremendous amount of his speeches here on Torah Anytime.
I have heard back from both kiruv organisers and the teenagers who needed that extra type of push that he has a successful approach that is appreciated.
But as Yaakov said in the comments, not everything is about proofs and convincing. Sometimes it's just a matter of showing someone appreciation and support while they find their own way through those turbulent years. This is something Rabbi Wallerstein is known for.
There is a great saying, apparently wrongly attributed to Samuel Clemens, which goes like this:
When I was a boy of fourteen, my father was so ignorant I could hardly stand to have the old man around. But when I got to be twenty-one, I was astonished at how much he had learned in seven years.
Kids will come to their senses eventually. Most kids, especially boys, have issues with authority in their teenage years. Unfortunately that restlessness translates into shucking off religious responsibilities too. Fortunately, they will come to their senses once they mature a bit. Luckily there is no kareis until 20, so by the time they come back there is little for them to worry about. However, by that time many feel they have no home or religion to return to, so they stay wherever they are religiously. That is our fault not theirs.
Also remember the Bartenura in Avos. Knowing what to answer to an apikores does not apply to a Jewish apikores, as arguing with them will only send them further off.
So, if I may offer a bit of advice, based on experience and the words of many people who spend a tremendous amount of time with children in these situations. The next time you see this teenager, instead of trying to prove God's existence and our duty to Him, just take the opportunity to say hi, how are you. This may take a tremendous amount of restraint on your part, but in most situations you will experience the results a few years from now.