Here's a breakdown:
* Ayarza: This part of the name likely refers to a specific location in the Basque Country. It might be an old village, a geographical feature, or a specific area within a larger town.
* Goitia: This part translates to "high place" in Basque. It's a common element in many Basque surnames and signifies an elevated location.
Therefore, the surname Ayarzagoitia likely means "from the high place of Ayarza".
To pinpoint the exact location, you would need to research the specific history of Basque surnames and delve into records of places called Ayarza.
Note: The spelling of Basque surnames can vary slightly depending on regional dialects and historical influences. Therefore, you might encounter variations like Aiarza, Ayarzagoytia, etc.