Irina is from the Greek Eirene (Εἰρήνη — peace) via the Russian Irina. Saint Irina (Irene of Rome, d. 304 CE) is one of the early Christian female martyrs venerated across Eastern and Western Christianity. Saint Irina the Empress (Eirene Sarantapechaina, 752-803 CE) was the Byzantine empress regnant (797-802) — the first woman to rule the Byzantine Empire alone, restoring icon veneration at the Second Council of Nicaea (787 CE). Central to Anton Chekhov's play Three Sisters (1900) where Irina is the youngest sister.
Irina 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 Irina reduce to 6, The Nurturer. This is a traditional interpretive system, not a factual claim about the name.