Syn is from the Old Norse Syn (denial, defense). Syn is one of the Asynjur goddesses attested in Snorri Sturluson's Prose Edda (13th c.) — Gylfaginning chapter 35. She guards the doors of halls against those who should not enter, and she is invoked as the goddess of denial in legal defense — the Norse divine witness in court cases where the accused declares innocence.
Syn 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 Syn reduce to 4, The Builder. This is a traditional interpretive system, not a factual claim about the name.