Years ago in high school, I searched through a Surnames book just for fun. It too said that my surname meant "owner of a small farm".
I have also contacted some european Alderete's, and have found on this web site that Alderete "originates" in the Asturias region, which is northernmost Spain.
This makes sense in a way, given that the "Christian"(and previously "Visigoth") Asturias region was pretty much not directly ruled by the Moors, who we know invaded up as far as Tours, France, before being repelled by Charles Martel and his armies.
I would not be surprised if there were Moorish settlers or intermixed offsrping of Moors in the Asturias region who "Spanishi-ized" their names later, especially as the Christian forces retook their Spain and expected their Moorish and Jewish neighbors to become "conversos", or converts to Christianity.
If this is the truth about our name, we should appreciate it and not be hisrotically selective or ashamed merely because we want to attach ourselves to some "glory" of Spain, which for 700 years was the glory of the Arabic/North African Moors.