Habits are hard to break, and the other person needs to have an incentive to stop eating meat and/or animal products. Much like New Years Resolutions, those "I'm thinking of" thoughts are just going to be dropped because there's no tangible motivation to follow through with them.

You can try convincing people by teaching them the health benefits from avoiding red meats, but realistically, you're not going to get far. There's a lot of misinformation and outdated research on the viability of vegetarian and vegan diets, and it's hard to change somebody's mind when they feel like it might be unhealthy.

This is why I'm hopeful for lab-grown meat being cheaper than actual meat. You're going to have the "GMO science evil" crowd that can't be helped, but the average consumer would gladly trade their ground beef for an equivalent-tasting alternative that saves them money. It's not vegetarian or vegan, but it solves the ethical issue of factory farming.