The 1870 census has a Chas. P. Beebe in Joliet, Illinois, same town as Eliza & her grandparents. Charles was age 39, born Connecticut, occupation - city collector, and living with a Mary Beebe, age 26, born Illinois.
Perhaps the elder Eliza died and Charles was unable to care for his daughter, so the grandparents took her in, and Charles later remarried.