Possibly the family needs thing (which is true for me now, but wasn't then). I'm not sure it would have even worked though. Once they bring up the prospect of a promotion, they've already used a bunch of political capital to open that spot for you and it's an insult to say no. I get is different for other companies or even just other departments, but that's how it was for that one.
The other thing is, my company measures your engagement and potential by how ambitious you are. The reason they wanted to get rid of me is because they thought with an answer like that I would be disengaged and didn't have any potential.
I don't know if it helps, but what I've done when turning down promotions is to be sure they know that I'm pleased and honored by the offer and value it. And that the reason I'm turning it down is because I feel I can provide the greatest value to the company in the position I'm currently in.
All of which is true, and phrases things so they know you're not turning your nose up at anything and that you have the company's best interests in your mind.
Seems like a good idea, but with the risk that if you say you're choosing to stay "for the company" then the company can insist that actually it's better for them if you go to the new role. If you say it's about your family situation, then it's harder for them to insist.
Well, that's the sort of risk assessment that only you can make. My experience has been that a company won't do that, but you know yours better than I do.
But if I were to work at a place where I felt similarly, I would absolutely be looking for a job elsewhere.