Righteousness in the context of the chivalry can best be described as a strong sense of right and wrong. It is about having a conscience and following it. On one level, morality was of course defined by Christianity, but in the highly legalistic societies of Western Europe it was also about "justice." Nobles and knights were the king's deputies and the executors of his justice. As such, knights were expected to enforce the law, contribute to the maintenance of law and order, and ensure justice in the abstract by recognizing and acting upon what was right and opposing that which was wrong/unjust -- even in the absence of specific laws and customs.
While it is hard to know the motives for actions, we can say with certainty that there is no recorded incident in which Balian d'Ibelin is known to have acted from base motives. He did not break treaties as Reynald de Chatillon did, nor did he usurp a crown as Guy de Lusignan did. He treated with Saladin as a devout Christian, never pretending (as Reginald de Sidon and Raymond de Tripoli allegedly did) that he wanted to convert. More important, there is positive evidence of Balian d'Ibelin's righteousness: first, his decision to take command of the defense of Jerusalem in 1187, and second the Treaty of Ramla.
In 1187, when Balian d'Ibelin came to Jerusalem on a safe-conduct from Saladin to remove his family to the comparative safety of Tyre, he could have chosen to stick to the letter of his agreement with Saladin. He could have remained only a single night and left the next day with his wife, children and household. He could have abandoned a city of roughly 20,000 inhabitants flooded with a further 60,000 refugees but without significant numbers of fighting men to the avowed vengeance of Saladin. That he instead agreed to take command of a hopeless defense -- although this meant that his own wife and children were put at risk -- strongly suggests that his sense of responsibility to others outweighed his personal desires. Perhaps that isn't the same thing as righteousness, but I think it is fair to equate the two. Balian had a choice and he chose not to do what was in his personal interest, but rather in the interest of others.
Again, when negotiating the Treaty of Ramla that ended the Third Crusade, Balian placed the interests of the majority of inhabitants ahead of his personal interests. In August 1192 Richard of England was determined to return to the West and the French were no longer taking orders from Richard in any case. In short, the crusade was disintegrating. Although Saladin's troops were tired and demoralized, the Sultan still held the better cards. Convincing him to sign a truce was certainly a diplomatic coup, and it is significant that Balian did not endanger the desperately-needed pause in the fighting by trying to regain his own barony of Ibelin -- although it was just 13 miles south of Jaffa, which Saladin granted to the Christians. The temptation to try to include it must have been almost unbearable, but rather than risk the peace Balian surrendered his own inheritance and source of wealth.