The answer is barba. Milady states the etymology directly: "the word barber is derived from the Latin word barba, meaning 'beard.'" It's a short, memorable fact — the entire profession's English name literally means "beard person," a reminder of how central beard care was to the original trade.
Tondere, tonsors, and queue — the other choices — are all genuine historical barbering-adjacent terms (tondere/tonsor relate to shearing or clipping, and "tonsorial" as an adjective for barbering comes from this same root), but "barba" specifically is the word Milady credits as the direct origin of "barber."
Milady Standard Barbering, 6th ed., Ch. 1 (The History of Barbering), p. 8: "the word barber is derived from the Latin word barba, meaning 'beard.'"
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More real History questions with sourced answers: What Position Did Tribal Barbers Hold in Early Cultures, What Does the Acronym NABBA Stand For, What Barbering Trend Returned for Young Men in 2010.
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