What or which one

Solution 1:

Swan in Practical English Usage (p622) says:

Which and what are often both possible, with little difference of meaning.

• Which/what is the hottest city in the world?

But he then notes:

We prefer which when we have a limited number of choices in mind.

• We've got white or brown bread. Which will you have?

I would add a further example from the nursery rhyme about the fish:

• Which finger did it bite?

Which is certainly the better choice here. 

What Swan does not mention, however, is that  What is... can be contracted (→ What's...), whereas Which is... cannot. For this reason, I suspect that What's...? is the more common choice in spoken language, notwithstanding Swan's second point above.

"What's... has also recently become much more common in written language, despite the necessity to search for the apostrophe on the keyboard. The nGram link shows a comparison of What's your favorite... and Which is your favorite....

What's your favorite / Which is your favourite *

Since there is a potentially infinite number of ice-cream flavours, I think "What's your favorite ice-cream?" is fine for an out-of-the blue question. 

But if you are standing in front of a rack with 5 different flavours in an Eiscafe , then "Which is your favorite ice-cream?   seems somewhat preferable.

And it also seems preferable in the context of asking Person B to specify which is the absolute favourite among the limited number of favourites they are referring to.

* Note: The nGram comparison works if you type in the strings directly, but the link above does not - presumably because of the apostrophe. I'd be grateful for advice on how to fix this problem.