i'd say a .380 hand gun with fragmenting bullets if the kids are in the next room separated by a thin wall and maybe a little insulation.
i think some people will disagree with this.
the reason i say this is because a .380 is strong enough to get the job done without serious over penetration, and fragmenting bullets are basically a ball of led shot compressed. when it hits a wall (or a bone) it fragments. you may have to unload more of them into your target to stop them, but even if you miss, hopefully they will reconsider and evacuate your home after they realize you're shooting at them.
just my opinion, but it really depends where the children are. if they are upstairs and you are downstairs, that's different. just don't shoot UP. and shoot whatever you see fit.
and if you're going to be gung ho about shooting on sight / sound, you probably need to train your children in what to do in case of a home invasion. also you need to tell them that once they are in their room, they should probably stay there through the night or risk getting shot......
or something to that affect. kidding about the last part.. but there IS SOME truth to it.
teach them NOT to come busting through your door at 4am, etc.
i'm incredibly jumpy when i'm asleep. any kind of disturbance wakes me up and sends me about 3 feet in the air... these days having lived with my girlfriend for 4 years, i don't immediately reach for a weapon or wake up throwing punches. i've trained myself not to do so when i hear her voice. but needless to say, she doesn't get anywhere near me when she has to wake me up for something.
the FIRST time she woke me up, she leaned over me and said "WAKE UP!" and literally got elbowed in the side of the head. of course i felt horrible, but at the same time, i was asleep, and that was strictly a reaction. but ever since then, she stands in the doorway to wake me up lol. i still jump though. anyway, yeah... if you're anything like that, be careful.
fortunately we don't have kids.