How to simmer bone broth safely with an overnight pause?

You definitely can't just leave it on the stove; that'd mean far longer than 2 hours in the danger zone. (See for example How do I know if food left at room temperature is still safe to eat?)

If cooking until it's done is out of the question, you need to try to chill it. Putting the whole pot in the fridge might not be the best approach, though; it could take quite a while to cool down. Even if that's less than 2 hours, you're going to go through another cycle later, and the time is cumulative.

The best thing you can do is probably to use an ice bath to help chill it faster.

Personally I side with your family here, I have a gas stove and the idea of leaving something on all night is not one I'm comfortable with. However, leaving the pot out overnight is not a good idea from a food safety perspective or a results perspective. You won't get a tasty result even if it is safe to eat.

A good option would be a low power electrical appliance like a slow cooker, these are designed to be safely left unattended for long periods. If you don't have one then your best option is to cool your stock as quick as you can. For this you will want to maximize the surface area - a shallow tray will cool much faster than a stockpot.

A way I've cooled things down quickly in the past is to take all my stainless steel flatware (sometimes even small pots) and put them into the freezer for 20 minutes, then put the frozen flatware into the pot. The flatware will absorb some of the heat and once they are hot - which takes just a few minutes - I take them out. If the contents are still too hot I will repeat the process. It means you gave to clean all your flatware but it is less cleanup than having to put stock in every small pan you have to cool.

According to this NYT article, it is safe to leave overnight with the stove turned off. In the morning, bring to a rolling boil for 10 minutes and then continue to simmer.

"What about my lazy method of letting stock cool overnight, then reboiling it first thing in the morning? Dr. Snyder gave it a pass because it would spend only a few hours below 135 degrees, not enough time for the bacterial spores to germinate, start growing and reach hazardous numbers."

Use the oven.

When I do this slow simmer thing I use the oven, not the stove top. It seems like even at lowest setting on the top, stuff always burns to the bottom of the pot. It is hard to get the heat low enough and there is no control over what temperature a burner produces in whatever is above it.

Put the pot in the oven at lowest setting. The nice thing about the oven is that it will keep itself at that temperature. It is like a slow cooker in that respect. I put a cookie sheet with a raised edge one shelf down to catch drips in case it boils over. Having the oven on 220 degrees all night might be more acceptable to your family than having the burner on all night.

If you are compelled to turn it off it will stay warmer in there than up on top. Leave the pot sealed; no final stir when you turn the oven off. If it got hot enough to simmer it should be sterile in there until you raise the lid.

This oven method makes great chili too.

If you're worried about food safety and have an oven which is considered safer by your family, you can put the pot in the oven heated to a temperature which is below the boiling point of water, but high enough to prevent bacterial growth. It would also give the broth a couple hours of sous-vide cooking.