Hello. The name Suastegui is basically Basque, but the curious thing about it is that the form does not really appear in Spain. People named Zuasti (or occasionally Zugasti) arrived in the Americas and in a number of places added the suffix to their name, becoming Suastegui. The change from Z to S was probably to keep the pronunciation. Z in much of Spain is pronounced 'th', while in Basque it is not. This happened in Mexico City, in Yucatan, in Sonora, in Cuba and in Ecuador. Probably other places as well. I don't know why they added the ending. The Zuasti of Spain originate in Navarra. The name was already Suastegui when it arrived in Teloloapan. People with the name Suastegui in these different places do not seem to be related to each other.