What does "bene bene dai" mean?
When I ask my italian friend "Come staì" she often responds with "bene bene dai".
I asked her to explain what it means and she said she can't really, it's just a common phrase that she didn't know how to explain.
I know that bene means well and that dai is a phrase that can mean "come on!" but literally translated "well, well, come on" doesn't make any sense. Can someone please explain it to me a little?

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DAI!!!!
Questo DAI a volte è una parola vuota. Si usa in tantissimi modi:

andiamo  dai! -> let's go, come on!
andiamo a bere un caffè dai! ->  a proposal

Sto bene dai = in  this case I would say that your friend is not 100% ok, otherwise she would simply  say " BENE!"
I would translate it in : I am ok, despite everything.

As  you might know, us italian are ALWAYS FINE ! it means that when they ask us how we are, we always answer BENE, and never MALE.  
Buono studio dell'italiano!
Irene

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The punctuation 
-- come va?
-- bene, bene, dài. > means, I'm still alive

*Dài is an equivalent of "come on" or "say" in spoken language, literally "Give it to me" so means "Let me say so". 
It's also a fast automatic answer when someone obtains a refuse to an embarassing question, or begs for mercy.

(The accent is used by accultured speakers to distinguish it from "dai" which marks the locative clause.)

Curiously, Polish and Russians also use the same expression as "Come on": 
--- Давай! Давай! -- have you ever heard that?

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It's an idiom, we can't translate it into English.

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That's great, Andrea! I guess "no complaints" or "can't complain" would be a pretty good translation, as that's a common expression for us too. That makes a lot of sense!

Grazie! :)

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I don't think that way of using "dai" has a translation in English. Personally when I say it  it's something similar to "nothing wrong happened" or "nothing to complain about" but again, it's very hard to translate.