Arduino for kids toy with Battery (safe), Can Arduino overheat the Battery with long running code
Probably the easiest/cheapest and safest option is to buy a original USB Power Bank and there you have it. Power banks for cellphones/tablets have overvoltage/overcharge/undercharge protection on them and they are also completely sealed, so it should be perfectly safe to handle one of those to your kids (Of couse not some cheap chinese crap). 

They can then charge the power bank over USB on any PC or Wall charger and then use it for hours/days.

If I'm allowed to give you a tip, you could eventually pick one of these here, they are pretty good in my opinion.

Of course there are a ton of others around, so you can just pick one which fit your needs.

On your case you just cut a USB cable and plug the ends on the Pro Mini. If a short is detected, Power banks often just shut off and only turn on when they are charged again (You might try this first). If you use a 3.3V Pro Mini, just put a voltage limiter between it. You might also write "3.3V" on the voltage limitor and "3.3V" on the 3.3V version of the Arduino and teach your kids that if the Arduino has "3.3V" written on it, they should put that "3.3V" voltage limitor before it (or just not use 3.3V variants for ease of use).