So Abu Bakr al-Baghdadi wants to be recognized as "Caliph Ibrahim." There seems no rush, even by fellow jihadis, to offer the bay‘a to him. It may be worth noting that previous Caliphs named Ibrahim have not been well-remembered; for the first, there is some dispute whether he was ever even accepted as Caliph; the second was the Ottoman Sultan known as Ibrahim the Mad. Not great precedents.
Ibrahim ibn al-Walid was named successor to his brother, the Umayyad Caliph Yazid III in AD 744, but was quickly opposed by his relative Marwan II and deposed. At most he ruled for a few weeks, and there's some dispute about whether he was ever fully recognized. Marwan deposed him but allowed him to live peaceably under his protection. Until the ‘Abbasid revolution of 750, when Ibrahim was killed with most members of the Umayyad house. Marwan II also was killed, after fleeing to Egypt.
Ottoman Sultan Ibrahim was Ottoman Sultan 1640-1648. He is known as Ibrahim I as Sultan, but if one accepts the Ottoman claim to the Caliphate, he would have been the second Caliph named Ibrahim. His reign was troubled to say the least, marked by family rivalries, executions, and provocations of neighboring powers. In 1648 the Janissaries revolted; Ibrahim's Grand Vizier was torn to shreds and Ibrahim deposed and executed. He is remembered as "Ibrahim the Mad."
Neither is a very inspiring role model.