Alkyone (Ἀλκυόνη) is the Greek word for "kingfisher." Daughter of Aeolus, wife of Ceyx — both transformed into kingfishers when she leapt into the sea on hearing of her husband's drowning. The "halcyon days" are the two weeks around the winter solstice when Aeolus calms the seas for her brooding.
The English word "halcyon" preserves her name.
Alkyone 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 Alkyone reduce to 2, The Peacemaker. This is a traditional interpretive system, not a factual claim about the name.