Cordoba probably became the capital of Muslim administration because during the conquest years, around 711-715, it did not surrender. It was conquered militarily. When cities surrendered, their current officials could work out a deal to send tribute; when they didn’t, they ended up directly ruled. That’s how it went with Cordoba, a former Roman town. It was centrally located in the southern region, and it required a military force and appointed Muslim governor, so it turned into an administrative center.
Abd al-Rahman found that controlling Cordoba allowed him to control most of the Muslim regions. Coming from the East, he had grander visions of what a capital city should look like. Its Roman arched bridge was a start, and as we know, he built a large new mosque on the site of an old cathedral and monastery. Each of his successors added to the mosque until it was fully four times larger, but still and always in a large square with an orange grove on one side.
Cordoba had been supplied with Roman aqueduct water since the first century. Roman engineers built stone-and-mortar tunnels to bring water from any springs in the region. Roman engineers had built so that the water pressure was kept even; when the water was conducted down a steeper hill, the aqueduct was smaller.
Water pressure coming down from springs in the hills allowed Cordoba to supply fountains. In the Middle Ages, fountains were the highest mark of civilization. Northern Europe had not yet learned how to create them either with natural water pressure or with cisterns. Travelers from Frankia and England were amazed at Cordoba’s Golden Fountain. Engineers under Muslim rule also re-routed the aqueducts to supply the Great Mosque with running water.
With water in apparently endless supply, Cordoba could also keep many parks and gardens. It was an island of green on a relatively arid plateau. It had two more urban features to surprise visitors: paved streets and public lighting. Roman roads of stone stretched all through France and parts of England, but the new Germanic Europeans had not yet learned to build their own. Their cities had only hard-packed dirt streets. Cordoba’s streets were cobbled some time between 800 and 900; Paris didn’t pave some streets until the late Middle Ages.
The al-Rahman dynasty wanted to rival Baghdad in every way, as ongoing revenge for the Abbasid overthrow of Umayyad rulers. So Cordoba had a House of Wisdom too, like Baghdad’s Persian-modeled study center. The two dynasties raced to accumulate the largest book collection; Cordoba also became a center of book copying and publishing. It’s possible that book copying was one of the few trades that a single woman, for example a widow, could fall back on. We know that at later times in medieval Europe, women who had been educated beyond the usual, by scholarly fathers, could make a living that way.
With book copying and libraries came other kinds of learning. Cordoba was the center of European scholarship between 800 and 1000. Muslims traveled freely between Cordoba and Eastern cities like Alexandria and Damascus, which in turn had travelers coming and going to places farther east. That’s how much of the technology and science of China, India and Persia reached Europe. Franks and other northern Germanic types, with names like Lothar, Otto and Conrad, could come to Cordoba or nearby Sicily for a time to acquire a graduate-school education in the latest learning from far off.
Starting in Abd al-Rahman I’s time, the emirs (later caliphs) built a private family residence about four miles away, modeled on the family palace compound of Umayyad Damascus. A town grew up around it, with suburban estates and villages between. It was called al-Zahra and became the capitol of the capitol, the place where Andalusian Caliphs received visiting kings and ambassadors. It was destroyed in the eventual downfall of the dynasty, around 1000.
Note: one of the interesting quirks of researching a topic like this is that a mainstream site, MuslimHeritage.org.uk, tends to attribute everything to the Muslim era. You have to read with a skeptical eye and challenge its assertions to get to the real Roman roots.