Damini is from the Sanskrit dāminī (दामिनी — lightning, flash of lightning). 4th century CE) by Kalidasa — widely considered the Sanskrit-language seasonal poem celebrating the iconic monsoon — where lightning (dāminī) is a poetic image for feminine beauty and dynamism. Princess Damini — Sanskrit-Hindu heritage naming.
Damini 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 Damini reduce to 5, The Seeker. This is a traditional interpretive system, not a factual claim about the name.