Madrid is a Spanish-Jewish name, and several people bearing that name were burned at the stake in the Inquisition. My direct ancestor, Alvaro de Madrid, met such a fate. There were several Madrid households among the Jews living in Toledo, Spain before 1492.
Where does the name come from? Beginning in 1391, Spanish Jews began to be persecuted in Spain. Many were forced to convert and were compelled to drop their original Jewish name for the name of a city, a landmark, etc. Probably, the Madrids of Toledo had been forced to convert but reverted to Judaism when the anti-Jewish tide diminished.
Many of the descendants of the forced converts often sought to emigrate to the Americas, seeking areas far from Inquisitorial control.