Ayn was chosen by Alisa Zinovyevna Rosenbaum (1905-1982) — born in St. Her libertarian-individualist philosophy of rational self-interest has had wide influence in American conservative and libertarian thought; advocates have included Alan Greenspan and Paul Ryan. Ayn Rand Institute founded after her death.
Ayn 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 Ayn reduce to 4, The Builder. This is a traditional interpretive system, not a factual claim about the name.