I would NEVER advertise a horse as "kid safe"...that is just opening yourself up to a liability lawsuit. I have horses that are very excellent and gentle with kids, but if I were to sell one of them, I would not advertise them as kid safe...
As far as what I look for and expect, I'm after a horse much like smrobs described. Well trained is always number one on my list of requirements, both on the ground and under saddle. Next is disposition and having some brains between their ears, a horse that can tolerate beginner mistakes and be confident in situations that a young/little rider may not be. A horse that needs an active leader to rely on in a potentially unnerving/new/horse eating plastic bag type of situation is not the one you want to toss a rookie kid on. Age is the next criteria though I have met and owned plenty of steady young horses that most would call 'kid safe' in controlled environments. Never would I buy a horse that only had a few years of training/riding time and with young horses that is what you get. I look for one that's had enough years under it's belt that it's had thousands of miles put on it, years of practical experience, been exposed to a lot of different situations and has seen the world a bit. That being said, being a senior aged horse doesn't necessarily make one a kids horse either. I've met and know horses into their late 20s that I'd never even consider putting a kid on. My 20 year old Hano mare is a prime example of that, she's a fantastic horse, very well trained but spent her life as an 'arena diva' and if you take her out of her comfort zone she is a lot of horse to handle. She's the "It's a stick, freak out" horse and needs an extremely confident rider to make up for her large lack of. She's getting better about it but no way would I let any child take her out of the arena.
Also agree with the point made about the particular child the horse is intended for. What I would look at for my daughter is much different than what I would look at for a new to horses family type horse. My daughter is only 7 but has been riding since she could walk and has a firm grasp of what buttons make a horse do what - ie: sidepass, spin, how to ride a spur trained horse, etc and has put it to practical use on a dozen different horses. Just got her a new mare last weekend, 17 yo QH mare that's been shown successfully in many disciplines from WP to reining to being a 19/20 second pole horse. She was not advertised as kid safe/broke because she isn't what most would think of when they hear that. She's not the old plug that will pack a kid around all day, she is sane & solid but still requires a rider not just a passenger. She did test my daughter on their first ride, (she wanted to gawk and see what was going on in the pastures instead of doing her atm job in the arena) with M being an assertive kid rider she got after her once and EZ went to work. Another child that would be afraid to get after her butt about misbehaving wouldn't have gotten anywhere with her.
For the very new beginner/kid/family type horse, I'd be after one like my old sorrel mare. She's the epitome of babysitter. She is one that only does what she thinks the kid on her is capable of. She is great to start them on. M started on her and was frustrated for awhile because Jana absolutely refused to lope for her. Her seat, legs, hands developed and once that happened the old lady didn't hesitate to give her what she asked for. That being said I still wouldn't advertise her as kid safe, even though it's been about 23 years since she bucked or did something stupid under saddle.