The name Cordoba is very relevant and it is the same city in Spain to which it refers. Whether it is a good or bad choice of name depends upon your point of view.

Spain was conquered by the Islamic armies, Cordoba in 711. It was the capital of a Caliphate and a major Islamic cultural center. It reached its height during the 10th and 11th centuries. Spain was slowly reconquered by the Christians from the north and Cordoba fell to them in 1236. Eventually all Muslims were expelled from Spain, as were Jews.

The great mosque of Cordoba was famous.

Now what you make of the choice of names depends. Moderate Muslims may look to that period as one of toleration and moderate Islam, with Muslims, Jews, and Christians living together and producing a vibrant, multicultural society. Conservative Christians have already seized on the name to claim that Islam is trying to conquer the West again, just as they did in Spain. Personally, I think that just as in selecting the site, this Muslim group has made an ill-advised choice and unnecessarily upset people.

Spain has not always been a catholic country (at least not completely). In 711 the Iberian Peninsula was occupied by Muslims, that remained there until the “Reconquista” in 1492. The territory under Muslim influence was known as “al-Ándalus”. The Arabian influence can still be observed in architecture throughout the south of Spain (and colonial architecture in Latin American cities), but also in culture and language, even in everyday life. For example, around 8 and 10% of the Spanish language is borrowed from Arabian, including many common words (especially those that begin with “al”, like “almohada”, “alcalde”, etc.).

http://upload.wikimedia.org/wikipedia/commons/5/59/Al-%C3%81ndalus_caligraf%C3%ADa_%C3%A1rabe.png

The name of Cordoba House is a direct reference to the Caliphate of Córdoba, which ruled over the Iberian Peninsula and a bit of North Africa, from 929 to 1031.

http://upload.wikimedia.org/wikipedia/commons/1/18/Califato_de_C%C3%B3rdoba-1000.png

This era is particularly known for tolerance, where Christians, Muslims and Jews coexisted peacefully, with freedom of religion and emphasizing the pursuit of knowledge. Some Greek texts that would become important later in Europe were kept and translated there, so it was also an incubator for the Renaissance.

As stated under “The Cordoba Legacy” in the project's website:

“While the rest of Europe slept in darkness during the 8th century, in Al-Andalus (Southern Spain) the Muslim Moors set about building a society that honored knowledge and fostered intellectual pursuits; gained strength from its religious and social pluralism, and subsequently became the inspiration for the rise of Western culture and institutions.
At the centre of this transformation was the city of Cordoba. From the development of early cosmopolitan architecture to the mosques that served as universities, Cordoban life was modern and sophisticated. From 900 to 1100, it was credited as being the largest city in the world, boasting over 200,000 homes, 600 mosques and 900 public baths.
The Muslims that governed southern Spain developed an inclusive administrative culture, and the face of public life was diverse. Christians served as administrators, governors and advisors to the Caliphs. Both Jews and Christians were able to practice their faiths with complete freedom and were granted the right to administer separate courts to uphold their biblical traditions and laws. The atmosphere of tolerance, inclusiveness and respect witnessed in Cordoba inspired erudition in the multiple religious traditions in Cordoba. Christian, Jewish and Muslim scholars gathered to translate and subsequently to revive the traditions of the classic Greek philosophers at a time when the rest of the world lay stagnate in feudalistic traditions and overpowering monarchies. Cordoba became the centre of Jewish intellectual endeavors, being home to some of the most influential poets and commentators of the middle ages, such as Judah Halevi, and Maimonides.”(1)