@supercat says "A nasty thing about liars is that when asked complex questions, they may answer in whatever fashion, other than 100% truthfulness, would be most vexing. That is a substantial complication which other answers fail to take into account, but the problem is still solvable as stated."
I agree! I suggest asking each girl one of the following two simple english questions:
"Who is the other girl with brown eyes?" or "What is your name?"
If you ask a girl with brown eyes "who is the other girl with brown eyes?", she will just give the right answer. What happens if you ask a blue-eyed girl this question? She will do something other than naming another brown-eyed girl.
"Alice, who is the other girl with brown eyes?"
If Alice points to Cindy, ask "Betty, who is the other girl with brown eyes?"
If Alice points to Betty, ask "Cindy, who is the other girl with brown eyes?"
If Alice points to Diana or Emily, you can ask either Betty or Cindy.
If two girls are now pointing to the same girl, or one girl is pointing to a second girl who is pointing to a third girl, then all three have blue eyes (because there are only two brown-eyed girls).
Otherwise the two girls are pointing at two other girls. One of the girls pointing, and the girl she is pointing to, have brown eyes. The other pair have blue eyes. The fifth girl, not pointing or pointed to, must have blue eyes.
You have one girl left to ask a question (Cindy or Betty, depending).
If she is the fifth girl, ask her "who is the other girl with brown eyes?" and she points to a blue-eyed girl, the other pair have brown eyes.
If someone is pointing at her, ask "what is your name?". If she says her name, she and the girl pointing at her have brown eyes, if she says something else, the other pair have brown eyes.
Now it is also possible that they would answer
"I don't know", "nobody else has brown eyes", "I am a fish", or whatever. These are also lies. Whoever does this has blue eyes.
So if Alice says "I don't know", then ask "Betty, who is the other girl with brown eyes?"
If Betty points to Alice, then she also has blue eyes.
Otherwise either Betty and the girl she pointed to have brown eyes, or, Betty and the girl she pointed to have blue eyes.
Then you can ask, "Cindy, what is your name?" to tell which is which.
But if Betty pointed to Alice, you can ask, "Cindy, who is the other girl with brown eyes?"
If Cindy points to Alice or Betty, she has blue eyes, and so Diana and Emily have brown eyes. Otherwise, she and the girl she points to have brown eyes.
Of course, Betty could also say "I don't know," which also tells us she has blue eyes, do the same thing as if she had pointed to Alice.
Same if Cindy says "I don't know."
Actually, I was not 100% truthful. If you look closely you'll see I have a different interpretation than @supercat.
So let me anticipate some objections. As I say, "who is the other girl with brown eyes" poses no issue for the brown-eyed girls. I imagine one could object that: if Alice has blue eyes, then, Alice is still lying if she names Betty, even if Betty does have brown eyes ... because Betty is merely "an" other girl. (Boo, hiss.) Okay, but we can pedantically reword the question to get around this. Like "Alice: out of Betty, Cindy, Diana, and Emily, name one girl with brown eyes". This is still something Alice would answer straightforwardly if she had brown eyes.
I think the actual two interpretation differences are,
(1) Is it a lie to give a correct answer when there are actually multiple correct answers?
(2) Is it a lie to say something which is obviously incorrect?
As demonstrated, it is possible to solve the puzzle even if liars might say "I am a fish" to any question.
Well, even if you disagree with my interpretation, hopefully you appreciate the solution anyway :-)