The grammatical rules for interrogative sentences are different from the rules that apply to simple declarative sentences. The Wikipedia entry for Interrogative gives examples from several languages.
The article distinguishes between polar questions that can be answered yes or no, and non-polar questions, where the questioner is asking for information, but leaves it to the listener to choose its form.
The form that questions take in a language is related to the social standing of the questioner and the responder, and the degree of politeness or deference that must be expressed, especially if the questioner is of lower rank. For example, the Japanese examples in the Wikipedia article are given in one of the polite forms. English has this too, e.g. when we ask or hint with expressions like do you think we should..., perhaps it would be better if... and so on.
The implicit question in most of these indirect cases is typically What do you think about X?, where X can be getting coffee, calling a doctor, and so on. The term what about is a particle. It forms a question out of the word or term that follows. It's a construction that exists in many languages.
In this case, the implicit subject is you. The implicit verb is think and the implicit object is the interrogative pronoun what. The expression about X modifies the verb think, so I suppose you could refer to it as an adverbial phrase. Note that "What about Starbucks?" would be just as logical as "What about you?", but it expresses something slightly different about the power of decision that the questioner is imputing to other party.
For children in ESL, how you explain the construction depends on the language they are coming from, and how questions are formed in their first languages. You may be closer to an answer than you think.