Mehrnaz is from the Persian mehr (sun, love, kindness) + nāz (grace, coquetry, charm) — "sun-graced." Mehrnaz in Persian tradition — appears in the iconic Shahnameh — Mehr is also the pre-Islamic Iranian deity (Mithra) whose cult spread throughout the ancient world. Mehrnaz Mohseni — Iranian-American academic. Princess Mehrnaz — modern Persian naming.
Mehrnaz 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 Mehrnaz reduce to 4, The Builder. This is a traditional interpretive system, not a factual claim about the name.