Delaney comes from the Irish surname Ó Dubhshláine — meaning "descendant of Dubhshláine" (the dark challenger; dubh = dark, sláine = defiance or challenge).
Delaney entered the U.S. top 1000 as a girls' name in 1993 and has been climbing steadily. Today it sits in the U.S. top 350.
Delaney 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 Delaney reduce to 3, The Communicator. This is a traditional interpretive system, not a factual claim about the name.