The commune name in Basque is Altzai-Altzabeheti-Zunharreta.

According to Jean-Baptiste Orpustan, the base (h)altz meaning "aulne" was used for the both toponyms Alcay and Alçabéhéty.
beheti means "at the bottom".

The name Sunharette comes from the Basque zunhar (Name from Soule meaning "elm" or "poplar") using the romanized locative suffix ette meaning the "place of elm".

The following table details the origins of the commune name and other names in the commune.

Sources:
Orpustan: Jean-Baptiste Orpustan,   New Basque Toponymy
Raymond: Topographic Dictionary of the Department of Basses-Pyrenees, 1863, on the page numbers indicated in the table. (
in French)
Cassini1: Alçabéhéty on the Ldh/EHESS/Cassini database
Cassini2: Sunharette on the Ldh/EHESS/Cassini database
Origins:
Duchesne: Duchesne collection volume CXIV
Ohix: Contracts retained by Ohix, Notary of Soule
Chronicles: Chronicles of Arthez-Lassalle
Soule: Custom of Soule
The commune name in Basque is Altzai-Altzabeheti-Zunharreta.
According to Jean-Baptiste Orpustan, the base (h)altz meaning "aulne" was used for the both toponyms Alcay and Alçabéhéty. beheti means "at the bottom".
The name Sunharette comes from the Basque zunhar (Name from Soule meaning "elm" or "poplar") using the romanized locative suffix ette meaning the "place of elm".
The following table details the origins of the commune name and other names in the commune.
Sources:
Orpustan: Jean-Baptiste Orpustan,   New Basque Toponymy
Raymond: Topographic Dictionary of the Department of Basses-Pyrenees, 1863, on the page numbers indicated in the table. (in French)
Cassini1: Alçabéhéty on the Ldh/EHESS/Cassini database
Cassini2: Sunharette on the Ldh/EHESS/Cassini database
Origins:
Duchesne: Duchesne collection volume CXIV
Ohix: Contracts retained by Ohix, Notary of Soule
Chronicles: Chronicles of Arthez-Lassalle
Soule: Custom of Soule
Alçay-Alçabéhéty-Sunharette