Florence comes from the Latin florens, meaning "flourishing" or "blooming" — from flos (flower). It is the English form of the Italian city Firenze, named for the flourishing flowers of its valley.
Florence Nightingale (1820-1910), founder of modern nursing, was named after Florence, Italy where she was born. The British singer Florence Welch of Florence + the Machine has kept the name visible. Today Florence is rising rapidly in the U.S. and sits in the top 350.
Florence does not currently appear in the US Social Security Administration's top 1,000 girls' names, so we don't publish a US rank or birth count for it. That says nothing about the name's standing elsewhere in the world — only that it sits outside the ranked US data we rely on.
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In Pythagorean numerology the letters of Florence reduce to 6, The Nurturer. This is a traditional interpretive system, not a factual claim about the name.