The problem with trying to convince someone of something is that they have to WANT to be convinced. You need to "sell" them on the reasons why they would EVER want to give up bread, pasta, desserts, rice....all that "good" stuff. Seriously, grains are such a part of most people's lives that they literally can't picture how they would manage without them.
Despite the fact that all the other posters have posted great (and quite valid) suggestions, I would think that the most effective elevator speech would address that person's own problems. What is it that THEY are looking for? Do they want their food to be "natural," or do they have digestive issues, immunity problems...do they want to have better breath, cleaner hair, fewer illnesses? Customize your "elevator speech" to their needs and you will have a 300% better chance of convincing them. Don't tell them how BAD wheat is. Tell them how GOOD life is without it (yes, I know that's not actually what you asked for). Realize that you are going to have to talk about this more than once.
Your first step should be to just get some interest going. I can't tell you how many people have zero interest in talking about food or health, and certainly not evolution :-) Keep coming back to the subject but just don't give the appearance of trying to push anything on them. Remember: in order to have any possibility for success, you have to make it attractive to them. And then, be ready for the fact that once they do it, they are going to think it was all their own idea :-)