You know, that's an interesting PoD you got there.
What effects would there be if Morphia of Melitene had given birth to a son instead?
Possibly a less clear succession : the military role of yerosolemite kingship was particularily stressed on, and if Baudouin II dies approximatly in the same time as IOTL, Baudouin's son would be too young to rule : I suspect you'd have at least some passing tought about making one of Baudouin daughters' husbands king of Jerusalem.
Indeed, the reign of Baudouin will be shakier, as one of the conditions of his return in 1125 after having been captured by Ortoqids was that he gave Yvette as an hostage : as he couldn't give another of his daughters (all married, and all tied to his political/diplomatical network) and that giving away his sole sone would be out of question...We might see the king not being released at this point before a longer time.
In this case, it could mean an earlier establishment of nobiliar domination in Jerusalem, especially out of the High Council. With Eustache of Sidon dead, Guillaume de Bures (Galilea) would probably reinforce his power as regent, possibly coming from trusted man of Baudouin IOTL, to main politcal ally or rival ITTL.
What would his name be? Baldwin, Hugh (for Baldwin II's father), or even Gabriel (for Morphia's father)?
I'd lean to Baudouin, giving it became sort of dynastic name at this point. Hugues as a second choice, maybe, but it would rather be to stress the ties with France but the House of Rethel wasn't that prestigious to really be referenced in naming in Jerusalem IMO.
If Baldwin II still dies in 1131, he'd be 11 years old. Who would be the regent?
Either Gillaume de Bures, or one of Baudouin's daughter husband IMO.
Melisende? Would she still be married to Fulk of Anjou
It's not really a given : she was married late IOTL because she was to inherit the kingdom, and that the need of Latin reinforcement was dire. ITTL, she might be married earlier, possibly to one of the Latin princes that were married to her sisters.
Maybe Josselin d'Edesse might end marrying Hodierne.