The High Court wanted Duke Hugh III of Burgundy, but he turned down the offer. Baldwin gave the King of France the authority to choose an alternate to Hugh, but since Sibylla already had a son who was in the line of succession through her, who would want to be king and see someone else's son inherit the throne?
According to contemporary chronicler William of Tyre, during the Holy Week of 1180, both Count Raymond III of Tripoli and Prince Bohemond III of Antioch marched to Jerusalem to force Baldwin to have Sibylla marry Baldwin of Ibelin and then give up his power immediately to his sister and new brother-in-law.
The Ibelin family had recently risen to prominence. Hugh of Ibelin (Baldwin's eldest brother), had become the third husband of Agnes of Courtenay, and Sibylla's mother. Sibylla's father, Amalric, after being annulled from Agnes of Courtenay, had married Maria Comnena. After Amalric's death, Maria Comnena had married Balian, the Lord of Ibelin and Baldwin's younger brother. The Ibelin family had become intertwined with the royal family of Jerusalem, then why not have a trifecta and have Sibylla marry an Ibelin?
Well, Bohemond and Raymond were foiled. Baldwin either knew of their plan and did not approve, or out of desperation he hatched his own plan.
A Poitevin knight, Guy of Lusignan, had recently come to Jerusalem. Guy was not the noblest of knights, having been exiled from Poitou because he and his brothers tried to kidnap Eleanor of Aquitaine for ransom. (This was a dumb idea.)
Another contemporary chronicler, Ernoul, records that Sibylla wrote to Baldwin of Ibelin to tell him that she would persuade her brother to let them be married on one condition. The condition was that Baldwin of Ibelin had to get himself out of captivity: he was currently held for ransom by Saladin, by whom he and Templar master Odo of Saint Amand had been captured in 1179. Ernoul, a squire of Balian of Ibelin, was probably saying this to make the Ibelins look better. Modern historians doubt Ernoul's account.
On the other hand, it is said that Saladin learned of the plan to force Sibylla to marry Baldwin and make him king, and Saladin raised the ransom from a knight's to a king's ransom, making it prohibitively expensive to get him released. (He was ransomed later by Byzantine Emperor Manuel I Comnenos.)
So Sibylla married Guy of Lusignan. Tomorrow we'll see how they handled the kingdom.