Halcyon (Ἀλκυών) comes from the Greek alkyōn, meaning "kingfisher." In Greek myth, Halcyone was transformed into a kingfisher with her husband Ceyx; the "halcyon days" of calm weather around the winter solstice are when she nests.
Halcyon is rare as a given name but distinctive.
Halcyon 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 Halcyon reduce to 6, The Nurturer. This is a traditional interpretive system, not a factual claim about the name.