I have not found any information about the surname AROIX, I only have found AROITZ, AROIZ, AROCH, I suposse than all of them are variations of the same surname, the meaning can be Gorge between mountains, or can be arotz, carpenter.

Basque surnames are names of houses or farms (baserriak), there are good works about the farms of Bizkaia and Gipuzkoa, but there are not works about the ones of Iparralde (French Basque Country).

Probably AROIX is a variation af other basque surname,

The only way that you have to obtain information is to look for his baptism record, You can go to a Family Research Center (In the mormon churches) and ask for the microfilms of the churches of Baiona (Bayonne). They have microfilmed church records all around the world. Bayonne is the main town of the French Basque Country (we call it North Basque Country)

As I have not information about this surname, I would be very pleased to know anything that you find , in order to include it in other books about surnames.