Esther is made into the housekeeper of Bleak House almost as soon as she arrives. It is, for her, an awesome responsibility, all the more so because she is only twenty. When she tells Mr. Jarndyce that she hopes she "is clever enough" to do the job, Jarndyce says,

"You are clever enough to be the good little woman of our lives here, my dear," he returned playfully; "the little old woman of the child's (I don't mean Skimpole's) rhyme:

"'Little old woman, and whither so high?'
 'To sweep the cobwebs out of the sky.'

"You will sweep them so neatly out of OUR sky in the course of your housekeeping, Esther, that one of these days we shall have to abandon the growlery and nail up the door."

The nicknames fix Esther's identity within the house as "the good little woman." This is meant as a complement, of course, but the effect of these names is to subvert or replace Esther's identity as an individual. As Esther says, 

This was the beginning of my being called Old Woman, and Little Old Woman, and Cobweb, and Mrs. Shipton, and Mother Hubbard, and Dame Durden, and so many names of that sort that my own name soon became quite lost among them.

So, in a way, calling the young Esther names like "old woman" or "Mother Hubbard" is remaking her into those things; her "guardian" Mr. Jarndyce is assigning her a role to play, and Esther hopes she is "clever" enough to play it well. Of course, Esther is more than clever enough to do so, and more clever than Jarndyce give her credit for being; one of the more interesting parts of the book is how Esther, as the narrator, is able to be what Mr. Jardyce expects her to be while at the same time retaining her own independent subjectivity.