Steven has a spoiler-laden discussion of Reki’s name that has some excellent screen grabs.
When Rakka gives Reki the box containing her true name, Reki opens it and finds a note from the Washii that tells of two stories, one for each name. The first story seemed to be of her pre-cocoon existence, whereas the second was of her existence as Haibane. But how can the Washii know so much detail about her previous life? And yet, if he was speaking of Reki’s possible fates in Glie, then the story was simply wrong; for example he says that Reki lost everyone she loved, and thus had to face her struggles alone. But she always had Kuramori – and then Nemu, and Rakka.
Update – here is the transcription from the last episode:
This is a story about a girl named Reki. She was doomed by an unfortunate fate, and even lost the people that she could share her sorrow with. Feeling herself worthless, she called herself Reki, using the letters which meant “small stone.” However, the letter that expresses her true name means, “the one who was run over and torn apart.”
after Rakka saves Reki:
REKI: Have I been.. forgiven?”
(Rakka sees that the Reki “small stone” name is on the stone, instead of the rekishi “run over” name)
RAKKA: I saw this break…
“If a bird brings you salvation, the name Reki will disappear, and the Reki that means stone will become your true name. With the belief that it will come true, I hereby introduce a new story of Reki, that is stone.
(and we see – Reki’s wings are gray.)