did laura wanted the "pendleton" to be rid off from her name or something or did i just re...

Yes, she's been saying that once her grandma dies, she wants to get rid of her last name (I assume that she assumes that her uncle will get rid of her. So I don't know if that's actually what she wants or if she's projecting because she knows that's what will happen) and then become a governess for a rich household who'll have her. I think that's also some of the reason why she's been carefully building up a lot of goodwill from the London crowd.