It's written with the kanjis of rei from gratitude and na from Nara. At first I thought that Rena was just the pronunciation of Reina but it has nothing to do with it. Then again I don't blame her for wanting to start over as Rena.

There are times in which a change of name is a good thing, I suppose.

[I do wonder if she will ever return to Reina. I suppose that once she's an adult and married she would, then again by then Rena would be even more of her name than it is already... It's not like she would need to change it again.]