Revealing that Flynn's real name is Eugene seemed a little bit out of the blue. I understand that his past as Eugene was very different than how he presents himself as Flynn Rider, but the whole name change that late into the movie felt out of place and added nothing to the story (at least for me). It has been a while since I've seen the film so my memory is hazy, but this has been something that has nagged at me since the first time I saw it!
Because "Flynn" is a persona he developed, a thief. Reverting back to his birth name indicates his dramatic change of heart -- due to falling in love with Rapunzel.
As described by Conny Eisfeld in her book How Fairy Tales Live Happily Ever After: (Analyzing) The Art of Adapting Fairy Tales:
In the end, Rapunzel does achieve a transformation of the formerly insincere Flynn towards his more humane Alter Ego Eugene as well.
His true name and backstory (i.e. being an orphan) were revealed about halfway through the movie. He revealed his name because he thought they were going to drown in the cave filling up with water (which made Rapunzel reciprocate with the info about her hair, giving her the idea that saved their lives). He shared his backstory because he trusted her after that, showing that he's started to change. It's also because he's starting to like her (as Rapunzel mentions right afterwards to Mother Gothel: "I think he likes me").
At the end of the movie, he mentions that he's going by Eugene again. This, along with the fact that he's stopped being a thief, shows that his transformation is complete. (And we see this is true in events that happen after the movie.)
Also, the first episode (transcript) of the animated series gave me the impression that he adopted more than just the name (well, that and Varian seems like an idiot):
Varian: I've read every single book about you!
Eugene: Oh well, um, you see, that's not actually me. I just took the name from the book.
Varian: Hey, hey! Remember the time you dueled that evil knight, blindfolded?
Eugene: No, no, not me.