Rajani is from the Hindi / Nepalese / Sanskrit रजनी (Rajanī — night). Rajani is one of the Hindi + Nepalese feminine names — central to traditional Hindu-Indian heritage. The Sanskrit rajanī (night) is central to Sanskrit classical literature Hindu cosmological + Vedic astronomical heritage.
Rajani 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 Rajani reduce to 8, The Visionary. This is a traditional interpretive system, not a factual claim about the name.