They are both nice, but Eliza would be more unusual. Agree the East London thing must be referring to 'My Fair Lady'- character of Eliza Dolittle who is a poor London flower seller - ie. they are saying it was a popular name with the Cockney working class in the 19 th century. But actually it was also popular with the upper classes around 1800, look in Jane Austen.
Don't think any of that applies now though so ignore!