As the title says, let's suppose that Godfrey for whatever reasons you want does not becomes the first ruler of Jerusalem (maybe he dies of illness, as he had suffered previously with sickness in the Levant; dies in battle, or maybe he simply packs his stuff and goes back to Europe, and so forth). Who can become the first ruler - even if does not picks the title "King" like Baldwin I did?
= Raymond of Toulouse? The Wiki says that he outright refused to be crowned, out of genuine piety, and, yet, his desire was to remain in the Holy Land, so he made an effort to wrestle Tripoli from the Arabs. Nevertheless, despite his refusal, perhaps we can imagine him being convinced by the secular and ecclesiastic authorities to adopt a neutral honorific, like Godfrey did (let's say, something like Prince of Galilee, Protector of the Holy Land, Defender of the Holy Sepulcre, or whatever), due to the real necessity of preserving the Holy Land against the Fatimids.
The advantage is that Raymond, like Godfrey, had heirs to be pointed out, like his minor son Alfons-Jordan, his elder (bastard, IIRC) son Bertrand (who was still in Europe, but IOTL claimed the County of Tripoli) or even his nephew William (who was in the Levant at the time and was likely to be known among the Crusaders).
= Bohemond of Taranto? If Raymond is not an option, I find likely that the other Crusaders might look to Bohemond. He didn't have the pedigree, neither the wealth, but his sheer charisma and apparent leadership of the expedition during the Siege of Antioch would count on his favor. As of yet, he wasn't a prisoner of the Turks, and Tancred, his nephew, wouldn't be the first choice. Now, a Norman Jerusalem could be an interesting scenario, especially if Bohemond preserves a territorial unity between Antioch and Jerusalem, it will be a much more formidable realm than OTL with the Boulogne Dynasty. Bohemond proved historically to be a very ambitious character, so I don't find it a stretch to assume that he might assume an invitation to become ruler of the Levant.
Also, due to his kinship with the Norman rulers of Sicily, Bohemond out of all the Crusader princes is the one that could, at least in theory, count on the support of the "closer" Mediterranean power in the face-off against the Egyptians and the Byzantines.
= Other minor noble? If none of the options above are possible, could we see another of the Crusader leaders being elected? Perhaps Robert Curthose, in spite of his poverty and the few number of men he committed to the expedition, he is still the son of the King of England, and a proved military leader. After him, what's the likelyhood of Robert of Flanders or even Tancred of Sicily being chosen? Perhaps in the end we might reach the same scenario of OTL, with Baldwin of Edessa becoming the very first king.
= An ecclesiastic authority? Now that Adhemar of Le Puy and Pope Urban II were already dead, I don't find this too likely, but perhaps Jerusalem could become an official theocracy with a de facto military rule by a lay prince. Possibly this arrangement might suit someone like Raymond of Toulouse or even Tancred.