In act 3, scene 1, when Tybalt tries to goad Romeo into a duel, Romeo resists, claiming to hold love for Tybalt himself. (Note, however, that this does not prevent Romeo from later taking up arms, spurred on by vengeance after Tybalt slays Mercutio.)
The meaning behind these sentiments seems to derive from Romeo's secret marriage to Juliet, Tybalt's cousin. Thus, through the institution of marriage, Tybalt and Romeo would be understood as kin, and (when seen in these terms), killing Tybalt would thus mean killing a member of his own family (an element that Romeo seems to recognize, even if no one else in this scene would be aware of these details).
In this sense, this scene has a very interesting subtext and dynamic at play, by which Romeo and Juliet's secret marriage has actually placed both Tybalt and Romeo in a deeply problematic situation, given Tybalt's bloodthirsty hatred against Romeo himself: should the two duel and one kill the other, the winner would then become a killer of family. That only Romeo is aware of this dynamic does not change the fact that it is in force.
Of course, in the end, this does not prevent Romeo from killing Tybalt, an outcome that proves disastrous for his marriage with Juliet. As punishment, Romeo is banished from Verona, and meanwhile, Juliet finds herself under increased pressure to marry Paris, circumstances that will result in the tragic conclusion of the play.