Claire is the French form of Clara — from the Latin clarus, meaning "clear," "bright," or "famous." Saint Clare of Assisi (1194-1253) was the founder of the Poor Clares, the female counterpart to the Franciscans.
Claire has been in the U.S. The single-syllable French spelling is the cleanest of the variants (Clara, Clare, Klara, Chiara). A short word for daughters who are meant to be unmistakably themselves.
Claire 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.
In Pythagorean numerology the letters of Claire reduce to 3, The Communicator. This is a traditional interpretive system, not a factual claim about the name.