Mnemosyne (Μνημοσύνη) is the Greek goddess of memory — from the Greek mnemē (memory).
The English word "mnemonic" derives from her name.
Mnemosyne 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 Mnemosyne reduce to 6, The Nurturer. This is a traditional interpretive system, not a factual claim about the name.