I would certainly be looking, but it is hard to stay what it would take to stay without knowing the specifics of other jobs you could realistically get.
The alternative at hand is about a 15% raise in total compensation, a reduction in daily commute time from 90 minutes by car to 45-60 minutes by bike (which is worth $5K alone in parking/gas/insurance), a bigger group of possible peers & mentors in terms of becoming a lead. On the downside, giving up what seems like a clear promotion, a product I'm probably slightly less interested in (but not totally bored by), child care costs would go up, possibly by an amount that ate most of the raise in base pay.
If I'm going to negotiate at all, it's going to be for a stupid large amount. But if they said yes to the stupid large amount, I mean, I should stay, right? So I've got to figure out how stupid to make it
I would take the new job for the commute alone. Child care costs may hurt a bit for a couple of years, but having that extra time and the health benefits of getting to bike to work will be so much better in the long term.
Given what you've said in your various responses, I don't think your friend will ever really be happy staying. So sure, ask for a 75% raise to stay if you want, but it doesn't seem really worth it.
Having worked in a totally dysfunctional place, leaving is a gift that can not be expressed in dollars. The world is a more magical place. The sun shines brighter. My coffee tastes better. I smile on my way to work and sing happy theme songs in the elevator.
And this place isn't that awesome - the old one was just sooooo bad. Don't stay.