Hi, I'm from Puerto Rico and in our family's research we found that Ponce de Leon was not really a name. It is beleived that Ponce was a title given to Juan after the battle against the Moors in Spain. He was the Ponce (somewhat like a Mayor) of the province of Leon, thus every time he signed his name he would sign it Juan Ponce de (of) Leon and stayed so to this day. He accompanied Columbus in his third voyage to America when Columbus discovered Puerto Rico in November 19, 1493 and was given the title of Governor of Puerto Rico by King Fernando and Queen Isabel of Spain. While in Puerto Rico he sent for his wife and three daughters to accompany him in the Island. Although he had no sons the name was not completely lost because his older daughter Maria in honor of her father after his untimely death asked her husband Juan Garrido (and he agreed to) to name their first born (a boy) with her maiden last name. That is how the name continued to this day.
Thought you may find it interesting!