This is a theory on how Elsa gained her ice powers. First of all, there has been quite a few hints in the movie that this takes place in the same world as Tangled, an equally popular Disney film. One of the most obvious includes that when Elsa is running from the castle in "For The First Time In Forever," You can catch a sneak peek of Rapunzel and Eugene walking towards the castle. And this is no mere misunderstanding. See for yourself.
This picture clearly shows Eugene and Rapunzel on their way to the coronation. And coming back to the topic of my theory. As many of the people who have watched Tangled recall, Rapunzel's mother was extremely sick during the time of pregnancy, and was given a medicine with the magic flower to heal her illness. Well, perhaps Elsa's mother came under a very familiar illness, perhaps that flower that had been used to heal Rapunzel's mother, is one of a kind of flowers that give special powers, now, the flower that healed her mother was a flower that held the powers of healing, perhaps, when Elsa's mother became ill, the soldiers went searching for something to cure the queen. Now, perhaps, this antidote, might have been in a flower of a similar kind, that's right, an ICE flower. Perhaps the queen had been under some sort of curse, in fact? And the only way to aid her was to find a certain flower. Now, when Elsa was born, her parents were shocked to see that she had been born with the lightest of platinum blonde hair, and when they brought her home, they were shocked to see that the moment she had touched the floor with her hands, the spot had frozen solid. They immediately took her to a doctor. But he had no idea what was wrong with her, all he recommended was that they keep her powers on they down-low, so that no-one would be afraid of her. But this curse was but a one-time consequence, none of their children from then-on would have it, hence Anna's lack of powers.