Melpomeni is from the Greek Μελπομένη (Melpoméni — she who sings, melodious one) — from Greek melpein (to sing/celebrate with song). Melpomeni is one of the Greek feminine names — central to traditional Greek mythological + Orthodox heritage. subject of pan-Hellenic dramatic heritage.
Melpomeni 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 Melpomeni reduce to 3, The Communicator. This is a traditional interpretive system, not a factual claim about the name.