Lillian is an elaboration of Lily — ultimately from the Latin lilium, the flower long associated in Christian tradition with the Virgin Mary and the quality of purity.
Lillian was popular in the early 20th century (peaking around 1900), declined for most of the 20th, and has returned beautifully since the 2000s. Today it sits in the U.S. top 75.
Lillian 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 Lillian reduce to 6, The Nurturer. This is a traditional interpretive system, not a factual claim about the name.