This is actually a really interesting topic to me, something we've struggled with a bit as well.
Sozlet was always thin to way too thin until she had her surgeries (adenoidectomy, tonsillectomy, etc.). I've since read about weight gain being linked to tonsillectomies. At any rate, in the six months or so after that she gradually gained weight, until I kind of had to admit she was overweight. (It was gradual enough, and I was so used to thinking of her as a stringbean, that it didn't really register at first.)
As part of my general anti-prejudice teachings, we'd had talks about fat. Not being judgmental, different reasons for people being overweight, being overweight doesn't necessarily mean being unhealthy, etc.
So, she was overweight, but was profoundly unconcerned about that.
I was though!
So that was super-tricky, trying to get her healthier without giving her new body issues.
I did focus almost entirely on exercise, but started talking about balance more. Not dieting (you're right, osso, re: my opinions there), but more "sure, you can have that, but have some fruit too." I was helped along a bit by the fact that her nausea issues are exacerbated by bad diet -- if she has too much fat/sugar/ salt when there are other risk factors (heat, exercise, dehydration, lack of sleep), she'll likely get nauseous. So I could put it in that context, but also just talked about healthy diets (usually in a more positive way -- this is good, not this is bad).
I also started talking about different kinds of fat. I have a bit more fat to lose to be at my ideal weight (though I've already lost a fair amount, and maintained that loss), and I've talked about that -- how belly fat specifically is linked to health issues. (I.e. someone who has flabby arms but not a lot of belly fat is healthier than someone with belly fat but thin arms.)
Then as she started losing weight (well, she didn't really, she just grew and didn't gain much more weight), I started saying that she looked healthier. Usually within the context of how strong/ fast/ etc. she was getting, too.
She's now pretty much right where she should be -- not too thin, and hardly any extra.
She seems to have gotten through this without any body issues. We'll see.
So tricky to navigate, though.
I read a thing semi-recently -- not sure if I remember enough to find it back -- about part of a parent's responsibility being to get our kids through to adulthood at a healthy weight.