Wow, I definitely want to follow this one -- my experience to date is that my kids seem to be largely hardwired, with one wanting All The Things, and the other who will happily pour out his piggy bank and count his money just for fun. So I think the lessons need to be targeted to what appeals to your kid; I just haven't totally figured it out yet.
E.g., I think DD prefers lazy. So I make her work for stuff; both kids get an allowance and a phone with data limits, but anything else she wants is on her. Because she strongly prefers playing to working, she is getting the experience she needs to be able to decide later the appropriate tradeoff between time to FIRE vs. stuff. I make her automatically deposit 1/3 of her allowance into a savings account, so she can learn the value of "invisible" money (because if she sees it, she will spend it, so "pay yourself first" is going to be a critical skill for her -- and she's definitely excited to see how much money she actually has in that "invisible" savings account after a few years). I have also focused on compound interest and such with her savings, and she has a small investment account so she can learn about that, but she is not an abstract/theoretical kind of kid, so that really hasn't made much of an impact. But it occurs to me that maybe I can talk about how that works both ways -- e.g., when you put extra money in savings or bonds or companies, other people are working to pay you; when you borrow money, you are working extra to pay other people. I tried this when she was younger (if she wanted an advance on her allowance, I gave her like 90% of it and charged her the other 10% as interest), but she always took the payday loan, so I thought, well, THAT's not the lesson I was going for and stopped. But now that she's a teenager she might understand it better, especially now that she has experience with her own savings account and investment.
And then maybe it's just me by example: I'm FI but still work, but that gives me a lot of flexibility, so I am working at home a lot, may well go back to part-time next year, etc. The parents are always the strongest influence in a kid's life, and they will notice what you do far more than what you say.
The one thing I did realize when DD was much younger is that a lot of this stuff can be invisible. I mean, when I shopped with my mom, I watched the money get counted out; when we went to the grocery store, I plugged everything in to my calculator to make sure we didn't exceed our food stamps. DD's experience, OTOH, is limited to watching the Magic Plastic Rectangle get whipped out and put away, and she doesn't see even that we actually PAY any money, much less that I am operating under a budget! I'm also very good at saying no to myself for things I don't need -- but again, that's all in my head. So I am thinking "boy that's cute, but I don't need a new top," repeated 10x, followed by "yes, I do need a new pair of grey pants for work, and these are a good price." And what she sees is "mom walks around, sees something she wants, and gets it." So I have learned to talk through my thought process out loud much more and to try to make the daily choices much more transparent. Again, not that that directly relates to FIRE; but it is exposing them to the constant choices that are involved in making that decision.
DS, OTOH, I don't even worry about. He is much, much more an abstract thinker, and naturally puts his money away until he sees something he really, really wants, and then is logical and intentional about it (he got a bunch of birthday money from relatives yesterday and spent about 40% of it on two things he has wanted from Amazon for several months (a nerf gun and a metal detector - he thinks he is going to find buried treasure, hah), but his savings account is still over $800 -- all from the allowance and other gifts that he did NOT spend). With him, all I'm going to have to do is show him the "Shockingly Simple Math" column, and he'll take it from there. :-) Of course, he's also the kind of kid who's going to want to work forever, as long as it can be in something fun like building robots -- he's like his dad, always has to have a project, and just loves tech and making stuff, so I could see him happily working in a lab or at a university basically until he keels over.
Tl;dr: As I think about it, most of my job is teaching the skills the kids will need to achieve FIRE if they decide they want to; the rest is just making sure they are aware that FIRE is an option that they can pursue if they want -- and that even if they think they want to keep working forever, being FI can sure make the ride a lot more pleasant.