Family get-togethers on Thanksgiving (or any holiday) may seem like a good opportunity to make your case, but I caution you against it. First of all, talking politics in those circumstances can easily ruin the event for everyone. And an argument certainly will. Second, persuading someone in a group situation is much more difficult than one-on-one (unless most of the people there are on your side of the argument). And third, many of your fellow infidels may be drinking alcohol, and that doesn't help with good listening or clear thinking.
The family gathering can, however, help our cause. You can use the occasion to observe and gain rapport. I suggest you focus this Thanksgiving on one person. Who is the most likely to be open to understanding Islam who will be attending the feast? Who is the most undecided? Pick one person.
During your family occasion, try to discover which representational system the person favors (click here if you don't know what that means).
And second, use your body to gain and maintain rapport throughout the day with everyone there, especially the person you picked (click here to find out how to do that). I suggest you do this at family gatherings of any kind.
These things will set you up beautifully for future one-on-one conversations with the person — conversations where you'll have a good chance of bringing them to a new understanding of Islam. In many ways, your task is half done when you are in strong rapport. Sometimes taking your focus off convincing and persuading can make you more convincing and persuasive. Sometimes not approaching something directly improves your ability.
I have seen a demonstration that perfectly illustrates this principle. In fact, I've done the demonstration myself several times after seeing it in a seminar. Here's how it goes: I toss something to someone, and they miss it. And they say something like, "I'm terrible at catching." So I tell them I'm going to test something. "I'm going to toss you this ball," I say, "but this time don't try to catch it. Instead, I want you to tell me which way the ball is spinning." Then I toss the ball, and to their great surprise, they catch it easily.
How does this work? They take their attention off trying to catch the ball, and instead pay close attention to the ball itself, and their body responds naturally and easily and catches it.
In the same way, if you take your attention off making people believe you, and instead pay close attention to their favored representational system and pay close attention to their body posture and match it, you have a good chance of making them believe you — easily and naturally — without even trying.
These two tasks are not time-consuming, difficult, upsetting, or conflict-creating. You can do them both and still fully enjoy the day too. Have a happy Thanksgiving.