The name Aldrete is of Spanish origin. Spanairds settledto New Spain, (Mexico) in the 1500's.
There is another family branch that spells the name Alderete. The Aldretes have been in the Mexico, Texas and the Southwest for many years - some say over 300 years. I am researching the Aldretes and have pushed my line back to the 1700's.
The Mormon family research center documents the Aldrete name going back to the 1500's in Valladolid, Spain. I hope I have helped some.