When Mrs. Higgins says that something walked in with her Elizabeth, she is talking about her future.
Later, in the same conversation, when Pickering and Higgins question the lady about what she meant, she explains that what walked in with her was "the problem of what is to be done with her afterwards." (i.e. once the bet is over, be it won or lost.) Mrs. Higgins worries about Elizabeth future, how she will make a living for example or what good the experiment will do her if she gains no money from it. (She yells to both men "The manners and habits that disqualify a fine lady from earning her own living without giving her a fine lady's income!")