You are almost certainly correct in your asumption, Robert. The Alderete family came from Tordesillas in Spain.(The Alderete tombs can be seen in their chapel) Tordesillas was long under Arabic control and was crucial in the reconquest of Spain. It is possible that the family had been Arab in background and had converted ( these peoples were called 'Moriscos'), although I cannot prove that. Either that or they took their name from some landmark with an Arabic name. Al- is certainly an Arabic prefix. It is like "the".
I might have an Alderete link. My grandmother was an Alterator. This was an Anglicised form of the name her great-grandfather from Chile had told to some official when he came to Australia in 1840. I suspect that it was originally Alderete.