Lila was born in Rome, Italy after her parents got remarried. Her name may have been a variation of Lilly, which is the name Grace was planning to give her first daughter after her great-grandmother and her favorite flower.
In a flash-forward scene in the series finale, she is shown as a college freshman who meets Ben, Will and Vince's son to whom she later becomes engaged. Their meeting parallels the relationship between Will and Grace, who also had dorms across the hall from each other.
They're not much, but they're mine. — about her boobs, The Finale