If you are referring to what Byakuya said to Zommari (the 7th Espada), Byakuya's pride is most likely Rukia, since in the anime when he was about to finish off the Seventh Espada, Zommari LeRoux, he said he wasn't killing him because it was his duty (as a Shinigami), but because he pointed his blade at his pride, since Zommari was about to kill Rukia before Byakuya got there. More generally, you can think of Byakuya's pride being his family - the Kuchiki Clan and all those who belong to it.