The LDS Family search site has the1881 Census
6 Ruabon St Birkenhead RG11/3574 folio158 page 71

Claude Quayle (it was mistranscribed as Quale) married 49 Labourer Iron Works b Maughold
Children in 1881
Mary 20 b Dumbarton Scotland
Isabella O 13 b Birkenhead
George 12 b Birkenhead
Jenette 10 b Birkenhead
James 13 b Birkenhead
Elizabeth 6 b Birkenhead
Claude 4 b Birkenhead
John 2 b Birkenhead
Wife Christina is not in 1881 census but was 27 in 1871 census and born in Scotland

In 1901 she was a widow in Birkenhead living with children, George and Claude.
As the family is missing in 1891, possibly they all went to America.

There is a birth for an Elizabeth Hester Quayle in Birkenhead in tht Jan-Feb-Mar quarter of 1875, volume 8a page 503.
With tis info you can order a birth certificate and find her mother's name