Eliza was born Elizabeth Lister at the home of her maternal grandparents, Philip and Susannah Pymble in Hoarwithy, Herefordshire when her seafaring parents were home from one of their many voyages to the colonies. Eliza, like her brother and sister, were often left with grandparents in Hoarwithy, or aunts and uncles London until her parents, after surviving a particularly disastrous journey in 1838, decided to settle in Sydney. Some time after 1841 Eliza with her brother John and sister Sarah travelled unaccompanied by family to Sydney to join their parents and siblings. Baby brother Henry had died of fever during the disastrous sea journey in 1838. In Sydney, the family lived in a town house facing the Domain next door to Parliament House. After a bankruptcy in 1849 they all moved to the country to manage a coaching house The Robin Hood and Little John at Mutton Falls. In spite of bushrangers and isolation they survived the first year. The following year 1850 Captain Lister died whilst returning from a shopping expedition to Bathurst. The widow Lister became the lessee of the Wellington Inn at Guyong and moved there with her large young family of six children. Eliza and her brothers and sisters helped her mother with running the Inn. They coped with the discovery of gold involving their old cabin-boy Edward Hardgraves and his deceit in swindling Eliza’s brother John and brother-in-law William Tom out of their reward for finding the first payable gold in New South Wales. In 1855 Eliza married Charles Arthur, the son of a family friend. After Charles died in 1866, Eliza married her neighbour Jack Bate in 1870. In 1872 Jack and Eliza took up farming at Vittoria near Bathurst.

Eliza died in 1899 and was survived by two daughters of her first marriage and Ellen Bate from her second.