Where does Elizabeth get her powers from?

Solution 1:

Potential spoilers below.

Rosalind says that it's probably because part of Elizabeth exists in another universe (her cut pinky), and thus can interfere with the tears/make new ones.

The actual quote: "What makes the girl different? I suspect it has less to do with what she is and more to do with what she's not. A small part of her remains from where she came. It would seem the universe does not like its peas mixed with its porridge."

As for the Luteces, when Fink sabotaged their tear machine to kill them, it somehow caused the siblings to be able to travel through tears freely, which is what they do throughout the game.

Solution 2:

Needless to say – SPOILERS!

In her Voxophone log, "The Source of Her Power", Rosalind Lutece says:

What makes the girl different? I suspect is has less to do with what she is, and rather more with what she is not. A small part of her remains from where she came. It would seem the universe does not like its peas mixed with its porridge.

At the end of the game, you will learn that Booker DeWitt sold his baby (Anna) to Comstock (who was assisted by Robert and Rosalind Lutece), to pay for his gambling debts. He tried to stop Comstock from taking his baby in the last minute, but failed to do so. As the tear closed, Anna's little finger was caught in the closing tear and was cut. Rosalind Lutece says that this is how Elizabeth (which is what Comstock named Anna as) got her powers.

^{This answer has been shamelessly copied from my answer in another question here.}