Very early in Frozen, we see that Anna's hair has a single streak of blonde through it, to signify the effects that young Elsa's accident has had on her memories of their childhood fun with her Ice Magic together.
At the end of the movie,
The undying love between the two sisters causes her to become completely unfrozen, and we even see the strand of hair is now red as it was in the beginning.
Does this mean that the character has regained their memories?
Anna doesn't regain her memories during the movie. Frozen's writer/director Jennifer Lee has confirmed this on Twitter:
Since "Elsa filled her in," Anna would know that her memory of skating with Elsa was a false one, even though she didn't have her old memories back.
Additionally, a follow-up book, Memory and Magic, is going to deal with Anna trying to get back her lost memories:
After years apart, Anna and Elsa are finally getting to know each other as sisters. But Anna still wishes she could remember the magical times they had together when they were younger—skating and making snowmen, even in the middle of summer! An eager young troll claims he can restore the memories that Pabbie removed, but Elsa's not so sure. Maybe it's time to make new memories!
Whether she'll regain her lost memories in the upcoming book remains to be seen.
Update: Memory and Magic is out now. It confirms that Anna doesn't get her memories back. A troll named Brock gives her a potion and tries to use a spell to restore them, but he isn't able to.
But Anna was still hopeful. Just because Brocks magic looked different didn't mean it didn't work.
"Now I'm going to ask you some questions, Just Anna," Brock said. "The answers will prove that my magic has worked!"
Anna nodded. Elsa looked at Brock expectantly.
"First question: what color is the sky?" Brock asked.
"Blue," Anna answered.
The troll looked very pleased. "What color is your hair?" he asked.
"Red," Anna replied easily.
"What does this have to do with anything?" Kristoff grumbled.
"I beg you not to interrupt, Book Crusher," Brock said. Kristoff opened his mouth to argue, but Elsa silenced him with a look.
Brock looked eagerly at Anna. "What is your favorite food?"
"Chocolate," Anna said.
"Aha!" Brock exclaimed. "Success! Your memory is saved, Just Anna!"
"Brock, you offered to restore my memories of Elsa's magic! From our childhood! None of those questions had anything to do with magic!" Anna said.
Brock the Mystical scratched his chin thoughtfully. "Oh, right," he said. "I knew I forgot something. Maybe you should cluck some more."
Anna's mind clouded with disappointment. Brock wasn't mystical. He certainly wasn't going to be powerful like Grand Pabbie. He was nothing more than a kooky little troll who liked to brew awful potions.
Anna's shoulders slumped. She glanced at Elsa and Kristoff, expecting them to say "I told you so." But they both looked just as disappointed as she felt. Anna realized suddenly that they both had been hoping it would work, too. They really cared about her. It made Anna feel just a little bit better.
Anna decides she can accept not getting her real memories back because she has Elsa to tell her what really happened.
After the smoke cleared, the trolls gave a hearty cheer. They were relieved that Queen Elsa had stopped the fire and that no one had been hurt. The forest was safe once again.
The trolls invited Brock to stay with them until his hut was rebuilt. Anna, Elsa, and Kristoff helped him gather his belongings from the rubble. The damage wasn't as bad as they had first thought. They were even able to save some of his moldy books.
Brock was still convinced that his potion had restored Anna's memories. Kristoff was eager to tell him the truth, but Anna didn't have the heart. Brock had tried his best to help her. The troll deserved to remember as much magic as he wanted to.
That night at the castle, Anna and Elsa sat down to dinner. Anna was excited to recall the day's events.
"I can't believe I actually clucked like a chicken!" she said.
"I can't believe you actually drank that awful potion," Elsa laughed. "The smell was horrible!"
Anna thought about how determined she had been to get her memories back. It had been important to her to remember on her own. Only that morning, she had felt like a puzzle with pieces missing. But this afternoon, all that had changed.
Anna realized that her memories weren't missing. As long as Elsa was there, the puzzle was complete. Elsa could remember the magic and Anna could remember the laughter. Together, they were a winning combination.
As @AlliterativeAlice has pointed out, a recent twitter discussion with Jennifer Lee, the film's Writer and Director has confirmed that Anna's memories have not been returned by the end of the film. In light of this, we can assume that the events in the final scene do prove that she must have been told that her memories were falsified.
Michelle Gray : Hello! I would like to know whether Anna got her original childhood memories back after she "thawed"? :-)
Jennifer Lee : No, they had been completely removed. Elsa filled her in, though, I'm sure.
Shortly after being struck in the head, Grand Pabbie converts Anna's memory of dancing in the hall (without skates) into a memory of her and Elsa ice-skating. The memory itself is fake which neatly sets up the final scene.
When Anna and Elsa are in the Courtyard in the final scene, Elsa creates a set of beautiful ice skates for Anna:
Elsa then waves her hand and magical ice skates (literally made of ice) form on Anna’s boots.
ANNA : What? Oh, Elsa, they’re beautiful, but you know I don’t ska[te]
Elsa grabs Anna’s hands and pulls her along on the ice. Anna slips and slides, but laughs in delight.
The only way Anna could know that Elsa knows they've never really been skating, is if she knows that her memories have been falsified.
The accident did not cause Anna's memory loss. The troll king removed Anna's memories of magic. There is no evidence that, after thawing, Anna regained those memories.
