Jin-gook:	연애야, 우리 뭐 먹을까? Yeon-ae, what should we eat?
Yeon-ae:	뭐 먹고 싶은 거 없어? Do you have something you want to eat?
Jin-gook:	너 먹고 싶은 거. Whatever you want to eat.

뭐 먹을까? (What should we eat?)

뭐 – an abbreviated form of 무어, meaning “what”; a pronoun used to refer to a fact or object that one does not know of
먹다 – to eat; have; consumer
~할까 – Proposition, the speaker asks the listener’s opinion about a possible future.

Casual – 뭐 먹을까?
Polite – 뭐 드실래요?

>> [뭐 먹을까?] can have different meanings depending on context.

>> Just as in the given context, if there is an emphasis on “뭐” meaning “what” the question would be asking what in particular the other person wants to eat.
A: 뭐 먹을까? = WHAT should we eat?
B: 난 김밥. = I want kimbap.

>> However, if there is an emphasis on “먹을까,” which is a proposition meaning “should we eat,” the expression can mean “should we have something to eat,” or “should we eat something.”
A: 뭐 먹을까? = SHOULD WE EAT something?
B: 그래, 나도 배고파. = Yes. I’m hungry, too.