"claim kin..."See in text(Phase the First: The Maiden - IV)
To "claim kin" in this context refers to a poor family asking their wealthy relatives to acknowledge them and help them financially. The Durbeyfields might therefore receive money if Tess is taken in by their d'Urberville relative. Tess would also rise in social status if she is acknowledged by a d'Urberville, and this (along with her beauty) would hopefully attract the attention of a higher-class husband (a "noble gentleman").
