For us, it would have to be something obscene, like over a $100,000 increase. I can't talk about it purely in financials, because there are so many emotional and social factors that affect how you view where you live. We live in a LCOLA with children next to DH's parents who provide free childcare. You can put a price on hourly childcare, but not on on the quality of young retired grandparents love.
Before we had kids, moving to Alabama was a difficult choice for me. I am originally from the Northeast and always imagined ending up in a city, so HCOLA. My job offers in California were a heck of a lot higher than what I started out making here and I moved away from all my family and friends, but in the end I think it turned out all for the best. DH's family is very closeknit and they are all locals, so we have a very strong support system. I made new friends which would have happened anyways since we had kids younger than most of my college friends, and I do think the LCOL made a huge difference to our savings. Yes, I would have made a lot more on the West Coast, but in comparing results with my cousins who live there and in other cities across the States and Canada, I think the move significantly increased my savings rate. In dollar terms, my West Coast cousins have more, but percentage-wise when looking at expenses, we are in a much better position. True, they could always retire and move from where they are now, but that is difficult leap to make all at once, though they talk about it every time they visit me and don't want to leave. We bought a house for $350k before the recession that is larger and nicer than my cousin's $1.5M house in the Bay Area. There are a lot of benefits to a smaller community, as well. We have a great bartering system with our coworkers and friends, so that means free eggs, venison, garden vegetables, pool use, condo stays, etc. in exchange for DH's computer skills and fishing.
However, there are some tradeoffs, like our education system is clearly not anything to be proud of, and there is less exposure to other cultures and ideologies, but we do a lot of supplementing at home and travel about once or twice a year. Maybe it is also possible to do what we have done in a higher COLA. Never in a million years would I have imagined I would end up where I am now 15 years ago, but I have been here nearly 10 years now, and it's like a dream. No regrets, and I am completely content. I have almost completed my metamorphosis from social party girl to at-home dinner party homebody.
TL;DR - We would move only if our new income allowed us to tough it out for 2-3 years before retiring fully and moving back home.