Lillian is having a remarkable revival, in the Top 25, the highest it's been since the 1920's. 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.