Get a candyfloss for Elizabeth

Hover over to see the solution (spoiler-alert):

Booker sold her to Comstock when she was a baby. Just before the portal closed, baby-Elizabeth stuck her pinky out of the portal reaching for her father (Booker) and it got cut off as it closed.

Elizabeth will give you things based on a few factors:

Your progress in the game. She will start throwing you things after you have a conversation with her about them. For instance, she starts throwing you money after Battleship Bay, when you encounter a vending machine. She'll mention that she's going to keep an eye out for money. Later, as you're heading towards Hall of Heroes, she will start throwing you weapons.

There's always a dialog the two of you have before she gains a new "ability" - she'll say something like "Those Vigors seem to be quite powerful, I'll keep you stocked in Salts" before she starts throwing you Salts in combat. I believe the unlock order is:

1. Money (Battleship Bay)
2. Health (Soldier's Field)
3. Ammo (Outside Hall of Heroes)
4. Salts (Inside Hall of Heroes)

(With thanks to hammar for jogging my memory about the precise order and locations on these)

The current situation and your combat strategy. If you're low on health, she'll toss a health kit. If you're low on ammo, she'll toss you a fully loaded weapon (always the same one I was using when she called out to me) and some extra ammo.

When she says she has something, she is at that point locked in to whatever she's holding. You can drop/change weapons, and she'll still throw you the one she's got - you'll drop your current weapon if you don't have one of whatever she's tossing. In my experience, health takes precedence, and then whatever combat action I was taking when she first started noticing I was low.

The difficulty and how many times in the current engagement she's helped you. For money, there's almost certainly triggers in the environment that count. You can leave the game idle, and she'll not throw you any money. However, it seemed like whenever I was near a shopping area with vending machines, she would toss me some extra money.

Usually she will notify you that she's "looking" for things when your ammo, salt, or health is about half, and then later she'll throw you something. This can be a good indication that it's time to change tactics - she'll reliably warn you that you're running low on ammo or salt, for instance.

In combat, you will only get so many items from her before she'll declare she's all out. When she declares she's out, you're out of luck. I've been down to my last clip in both weapons, a sliver of health, and no salts to speak of, and she'll just cower in the corner. At that point, it's skyhook time.

I've found that at harder difficulties she can't find as much stuff.

Some other notes about Elizabeth's game mechanics:

• When you're catching, you're invincible. It's still a good idea to take cover prior to catching, since you're likely to be somewhat disoriented as the camera angle switches around a couple of times in quick succession.
• She can throw through solid objects and at infinite distance. She has a way of teleporting close to you when she has something, but I've still managed to get her to throw through walls at times.
• You can't catch things while aiming down sights. If you're low on health and in a combat situation, this is a good thing to keep in mind.
• The items all appear to be objects that are not in the environment. She's not grabbing a fixed spawn jar of salts, or looting a corpse, she's conjuring one out of thin air.
• The implication of being thrown a fully-loaded weapon is that you can bypass the reload delay on a weapon by firing until empty, and then accepting her assistance. This can be helpful in some "last ditch" combat situations, since you trade the reload delay for a period of invulnerability+free reload+free ammo.

Elizabeth can also open rifts to give you a combat advantage (this starts in the Hall of Heroes), but that's another topic... :)