To trigger this you must have Lily get low health 3-5 times while fighting something.
All I did was allow her to fight stuff and not help her and once she got low health and the steps trigger for the quest, then heal her back up with stimpacks.
Just do this 3-5 times and then wait a bit and you should hear the recording of her talking with children and that is when you talk with her and can finish the quest and tell her how to take her medicine.
So I've been trying just this, and I've done it at least five times now - gotten her to low enough health in combat that she has her "psychotic break", then waited for it to end, then doing it again. But still, nothing appears to be happening afterwards.
Another thing - I'm not getting any popup messages for her phychotic breaks (apart from for her very first one). I know she's having them because I have her constantly set to "passive", and she becomes "aggressive" and I can't change her back for a while. And then there's the audio clues - her aggressive roar when she enters the break, and comments like "Oh, did I do all that?" and "Gracious me, I lost control" when it ends. Am I supposed to be getting popup messages for it, or is this normal?