Damayanti (दमयन्ती) is from the Sanskrit damayati (to subdue, to tame).
Featured in the Mahabharata, Vana Parva.
Damayanti 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 Damayanti reduce to 7, The Seeker. This is a traditional interpretive system, not a factual claim about the name.