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What do your friends call you?
Literal
Friends [topic-は] you-of matter [object-を] what call-[polite-のですか]?
A polite question of the form 'by what [name] do your friends call you?' なんて ('what, what kind of') is the casual interrogative, contracted from 何と. The のこと ('about, regarding') softens the question, making it about your identity as a person rather than just your literal label. Polite but conversational.