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You could be right with the Basque idea, but this may be a Hispanicised form. Go to www.buber.net/Basque/Surname and click on the name BEITIA. See what you think.
The Latin suggestion is probably a red herring. Avitius might be a Roman clan name or a cognomen, I couldn't say. Avitia would be the form taken by a female. I doubt that it would survive as a surname, maybe as a female given name, like Antonia.