I certainly don't have any magic recipe to change another person's mind, but the most effective argument I've heard is a direct personal question: "If it's a choice, then can you choose who you're attracted to?".
The first time I heard that, it resonated with me.
My big brother is gay. When we were teenagers, he had Playboys, and I never considered the possibility he way gay. I found out when I was in my early 20's. I only found out because I wanted to move in with him, and he decided to tell me, because otherwise he'd have to say "no". So, no big deal, I didn't care and I moved in. And so did his boyfriend, which was the reason he needed to tell me and find out my reaction.
But I assumed it was indeed a choice - that he had been straight as a teenager and decided to be gay later, rather than discovering it or admitting it. It wasn't really a clear thought, because I didn't actively wonder about it. But later on, I heard other (straight) people debating that, and I asked him. He told me it's not a choice, and that simple question - whether I could decide to be attracted to men instead of women - certainly clarified the question in my mind. No, I couldn't. So why should I think that he could?