Lillian is having a remarkable revival, rising to a peak of Number 21 in 2010 (the highest it's been since the 1920's) before dipping slightly to Number 28 last year. It was a Top 10 name in its Lillian Gish-Lillian (born Helen Louise) Russell-Floradora Girl heyday at the turn of the last century.
Now, with its trendy multi-L sounds, Lillian is seen by parents as a more serious and subdued cousin of the megapopular Lily. Lilian is another spelling.
TV actress Mary McCormack and director Baz Luhrmann both have daughters named Lillian.
Lillian probably originated as as a pet form of Elizabeth, completely independent of Lily.