Hispanicnames.com says it's Mozarabic
I found a site at Hispanicnames.com
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If you click on "History of Names" and search the page for "Alderete", it claims that the name is Mozarabic (from the Moors who invaded) and means "small farm".
I once saw this long ago in an old library book when I was looking up our name just for fun and didnt really take it that seriously.