I followed the two links Mr. B Alderete place in his ruminations. The "http://www.hispanicnames.com" and "http://www.s-gabriel.org/names/juliana/isabella/MoslemFullNames.html". The first one is now an extinct domain, the second one said the the Alderete surname was probably Arabic based upon the name. This sounds like it was guess work.
The Arabic word for farm is "????? ". I'll try to spell it phonetically in English alphabet,"mazraala". That does not sound like al derete to me. I tried to see if perhaps it was the Berber word for farm. Unfortunately there are no dictionaries for the Berber language.
As you know the Spanish word for farm is "granja". It seems to me if they were Arabic they would have changed their name to "lagranja" if they wanted to prove purity of blood. It all seems like guess work to me or filing a square peg to fit in a round hole. We need more research.