This achievement is earned from bringing a child back from the mood state of "Broken" back to the mood state of "Normal".
Broken is the lowest mood state a character can achieve, below is the descending order of mood states characters can go:
Normal
Sad
Depressed
Broken
Getting a child to broken mood state isnt fun at all. You may have to purposely treat the child badly to get them there. For example dont let them sleep in a bed, dont comfort them, keep them hungry etc ...
Its terrible, I know. Not cool at all.
A more natural way of getting them to this state is to let them react to something that happens around them (and it will) as the days go by. Children get sad and depressed quite easily in this game. So when they are at the depressed level, do not comfort them for a few turns and they should reach the broken stage.
Once the child (or any other character for that matter) reaches broken state, act on cheering them up immediately. This includes talking to another character, making sure they are fed, making sure the house is warm if its during winter, doing good deeds for strangers and that they have a bed to sleep in at night. The child can also play with toys around the house, yet may not want to until they have gone from "Broken" to "Depressed" and from there its pretty easy to get them back to a normal state again, and after this the achievement will pop.
Should also mention that the mood of your child affects the other people in the house in a big way. So if your child is in a negative mood for a long period of time, your other characters will start to have negative mood consequences as well!
Good luck guys.
This achievement sucks ...