Fallout: New Vegas Walkthrough - Lily's Companion Quest
Lily has a very brief companion quest that allows you to either upgrade her combat abilities or help her reach a much more clear mental state. If you'd like to advance Lily's story more, just look at this short guide.
Lily - Recruitment
Lily is one of the easier companions that you can recruit. Once you make it to Jacobstown, you just have to start up the quest "Guess Who I Saw Today." She'll be available to join you as soon as that starts. If you forget to pick her up during the quest, you can ask to join you on your travels at any point later in the game. She'll be outside tending the brahmin pens.
Lily is a fairly useful companion. She's not too good with her weak assault carbine, but her vertibird sword is incredibly useful at close range. She's also built like a tank and can even take a deathclaw one on one. If you want someone who can just wade into a battle and come out surrounded by bodies, she's a good choice.
Also note that Lily is amoral. She doesn't care about what factions you support and doesn't seem to be aware of the evil acts you do (Leo would probably support them anyway). If you want to do an evil run or a Caesar's Legion run, then she'll be one of the few companions that won't leave.
Lily's Companion Quest
Lily's quest isn't actually that complicated. You basically just have to travel with her for a bit and pay attention.
Lily has three important triggers for her quest. You'll need to see her enter a psychotic rage, see her take some medicine and hear her children's recording.
The first trigger is actually a little tricky if you're a good fighter. Lily needs to be at low health before she'll enter a psychotic rage. Start a fight with a big force and let Lily soak up a bit of damage on her own. You should get a note or que of some kind that she's entered a psychotic rage. After the fight is over, you'll have to wait a moment for her to calm down enough to be able to talk to you. Once that happens, you can ask about the rage. She'll explain Leo's influence over her and give you the first trigger.
For the second trigger, it seems like you just basically have to travel with her for a little bit. At some point, she should just take some pills in front of you. There will be a brief note from Lily as she calls out that she has to take her medicine.
The final trigger should happen shortly after that. If Lily hasn't played the recording of her children around you yet, then you should probably try sleeping. Waiting doesn't seem to trigger much, but actually sleeping in any bed in the wasteland should trigger Lily to play the recording in front of you.
Helping or Hurting Lily
After you see all three triggers, you'll be able to talk to Lily about her medicine and her children. She should explain that she's only taking a half dose so that she can remember her grandchildren. You have a big decision to make now.
If you want, you can just tell her to stay the course. She'll be able to remember her children, but will still be prone to psychotic breaks and hear Leo.
If you want to genuinely help her, you can tell her to take the full dose. As she fears, this makes it hard for her to remember her grandchildren, but she'll be much more calm. There also seems to be a major hit to her combat abilities, so keep that in mind. Since it only really has story benefits, you will probably want to wait to make this decision until near the end.
You can also have her stop taking the medicine all together. This has the result you would expect for the story. This gives her some major combat bonuses, but causes her to enter psychotic rages more often. She should now enter one from merely hitting half health.