Eulalie is the French form of Eulalia — from the Greek eu (well) + lalein (to speak) — sweetly-speaking, eloquent. Saint Eulalia of Mérida (c. Saint Eulalia of Barcelona (c. 290-303) — separate martyr; Barcelona's primary patron saint; the cathedral of Barcelona (Santa Eulàlia) was named in her honor. Eulalie (1845) — Edgar Allan Poe's poem; the title character is a personification of romantic joy; written shortly after The Raven.
Eulalie 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 Eulalie reduce to 2, The Peacemaker. This is a traditional interpretive system, not a factual claim about the name.