Why did the descendants of Ishmael (as) receive prophethood?

For the same reasons as the descendants of Isaac (as). Whoever understands why God gave prophethood to one line of Abraham (as) also understands why He gave it to the other.

Many people readily understand why the Children of Israel, the descendants of Isaac (as), received prophets. With the descendants of Ishmael (as) they then wonder what is supposed to be different. But one cannot build an upper floor before the ground floor stands. Whoever first works out why one line received prophethood already holds the answer for the other.

The first verse above shows the standard. God promised Abraham (as) to make him a leader of men. When Abraham (as) asked about his offspring, the answer was that this covenant does not embrace the unjust. Prophethood, then, is no inheritance received by descent alone. It follows faith, obedience and righteousness.

This standard applies equally to both lines. Both sons were sons of the same father, and both grew up in the same faith in the One God. The second verse above says that God placed prophethood among the descendants of Abraham (as), without excluding either line.

For the descendants of Ishmael (as) something more is added. Abraham (as) and Ishmael (as) together raised the foundations of the House of God in Mecca, and as they did so they prayed that God would raise a Messenger from among their offspring, as the third verse above shows. That prayer was fulfilled in the Holy Prophet Muhammad (saw).

The values, then, are the same. Devotion to God, obedience and righteousness led one line to prophethood just as they led the other.