It takes roughly an hour to drive from Khiva to Biruniy and a further hour to get to Ayaz

The Legend of Ayaz Khan
The origin of the name Ayaz remains uncertain. 

According to Sergey Tolstov, the name Ayaz comes from a Qazaq legend about the origin of Khorezm and the Aral Sea. A long time ago the lands below the current Aral Sea were occupied by two tribes, the Adagy ruled by Fasyl Khan and the Baysyn ruled by Ayaz Khan. Ayaz Khan had once been a slave but thanks to his wisdom and justice he had eventually become the leader of his people. The Amu Darya and Syr Darya at that time flowed into the Caspian Sea. But when Fasyl Khan dishonoured the daughter of a religious mullah, the course of the rivers changed and as a punishment flooded the lands and towns of the Adagy below the surface of the Aral Sea. To escape the flood, Ayaz Khan and his people moved south and so founded the kingdom of Khorezm. 

Links
Those interested in the conservation and architectural details of the Ayaz Qala sites will find the following report on a recent restoration project particularly useful: http://unesdoc.unesco.org/images/0014/001482/148218m.pdf.