Flannery is from the Irish surname Ó Flannghaile — flann (red) + gal (valor).
Subject of Brad Gooch's Flannery: A Life (2009).
Flannery 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 Flannery reduce to 5, The Seeker. This is a traditional interpretive system, not a factual claim about the name.