Why was the Arabian horse developed?

The Arabian horse is called such because it was discovered in Arabia.

Technically the Arabian horse came from Iran, not Arabian, but they were transported to that area by nomadic tribes. Up until about the late 1800's any horse that came from the Middle East, North Africa, or certain parts of Asia were all referred to as Arabians. During the formation of the Thoroughbred, many horses were wrongly called Arabians even though they were Barbs or Akhal - Teke's due to the location of where they were bought at.