POD:
October 976 - Caliph
Al-Hakam II does not die, instead living an additional 20 years. IOTL, Al Hakam II died at the age of 61. His general, who would later be called
Al-Mansur, placed a 10 year old boy on the throne and then imprisoned him, essentially ruling in his stead. Al Mansur was a great general, but not as enlightened a ruler as Al-Hakam II, and his actions led to bloody civil war, and in just a few short decades the nation shattered into over a dozen little states. THIS is when the reconquista actually began to succeed IOTL.
In this ATL, Al Hakam II lives an additional 20 years, dying at the respectable age of 81. The man who would have become Al-Mansur is instead kept in check fighting the enemies of Cordoba, eventually conquering entirely the Christian lands south of the Pyrenees (he was close to doing this OTL). Instead of a boy imprisioned all his life, the heir at the time of Al-Hakam II's death is a mature but vigorous man of 30 years raised in the traditions of Al-Hakam II. This provides a more secure base for the development of the Andalusian Empire in the centuries to come.
Ibn al-Haytham is considered by many to be the true father of the scientific method and modern style experimentism. He really was in Cordoba in this time period. What happened in OTL is that most of his work was underappreciated in the West due to the fact that the states in Iberia were a wreck and it wasn't until the reconquista conquered areas with his books that his teachings began to be spread once more. ITTL, he has a longer time under the patronage of Al-Hakam II, and his successor. His teachings and methods become widespread in Andalusia, and later spread to the surrounding nations that send people to Andalusia to learn. While this doesn't cause an overall faster rate of development after this, it does give everything a bit of a jump-start.
Roger Bacon was a slightly later promulgator of these thoughts, about 200 years later in fact, so instead of waiting for Roger it begins 200 years earlier.
History:
The History of Al-Andalus is a long and respected one which can be traced back to the founding of the Emirate of Cordoba by the last scions of the Umayyad Caliphate. In the 900s the Emir of Cordoba declared a new Caliphate, and was referred to in early texts as the Caliphate of Cordoba. The first Golden Age of Al-Andalus began with the reign of Caliph Al-Hakam II (born January 13, 915, died December 25, 996). At the time Al-Andalus was balanced on the edge of greatness or destruction. Under Al-Hakam II Andalusia was probably the most advanced nation in the world. Cordoba had overtaken Constantinople in size. The Moroccan region of North Africa was pacified and the Christian states forced into a small strip of the North of Iberia, and had had to acknowledge Andalusian dominance. Under the reign of Al-Hakam II the Caliphate grew with the final defeat of the Zirids and Fatimids in Northwest Africa and the full pacification of the Christian lands up to the Pyrenees under the able generals Ghalib and
Ibn Abi Amir. Al-Hakam II brought many books to be translated in Cordoba, including many first additions, and was a patron of the founder of modern scientific thought, Ibn al-Haytham. By the turn of the millenium, Andalusian influence had spread to Tunis and Sardinia, and thence the Emirate of Sicily where Andalusian intervention ended the internecine strife between claimants to the Sicilian Emirate in return for fealty to the Caliphate.
Rest of history needs expansion, but basically due to Al-Andalus holding together the advance of knowledge is about 200 years ahead of our own. Andalusia was one of the pioneers of the age of exploration, following fast on the heels of the French after their discovery of the New World and the Aztecs, such that most of Meso-America and South America practice Islam (Andalusian explorers converted leaders of client tribes to Islam and overthrew the Inca, and formed outright colonies in the less populated Central America, as well as the creation of a Canal linking the San Juan River and Lake Nicaragua via a canal through the Isthmus of Rivas to the Pacific. The Andalusians also revamped the old Egyptian Canal and created a new one through Suez to facilitate travel between the Mediterranean and Red Sea (especially helpful for the Hajj). Andalusians eventually colonized the Malaysian/Indonesian region, Australia and New Zealand.