Liz is the English short form of Elizabeth — from the Hebrew Elisheba ("my God is my oath"). "Liz" specifically — distinct from Beth, Lisa, or Eliza — was popularized in the 1950s-1960s, especially through Elizabeth Taylor's nickname.
Liz 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 Liz reduce to 2, The Peacemaker. This is a traditional interpretive system, not a factual claim about the name.