What a good, tough question.
I'd suspect that it's less of an absolute "magic number" and more like a ratio compared to the alternative. Like a 2X or 3X salary increase, maybe more.
If I was making $30K a year (slightly less than the national individual median income) and had kids, I'd have to strongly consider a $150K job that could open up a completely different set of opportunities for the kids. In a lot of places in America that's a leap from "borderline poor nutrition and you better hope you're smart enough to earn a full college scholarship... and try not to get shot in the meantime" to "safety, healthy food, and we're able to invest in your future."
If I was already making $125K a year that $150K wouldn't be so persuasive. Not upending my life for that.
Although, I suppose, once we start talking nannies and tutors we're probably looking at closer to $250-$500K anyway. A full-time nanny is... well, a full-time employee.