Alaia is a Basque female name meaning “joyful, happy” from Basque alai (joyous, happy) which either derives from Latin alacer (lively, brisk; glad, happy, cheerful) via an uncertain origin; or it cold have been borrowed from Spanish alhaja “jewel; gem” which ultimately derives from an Arabic origin.
Spelled Alaïa it’s a French surname, the French form of Alia(an Arabic female name “exalted, loft, high, sublime”), and Alaia is also an Italian surname though I’m not sure if it also derives from Arabic or a different source entirely.
Alaia is also a variant spelling of Alia/Aliyah, an Arabic female name, the feminine form of Ali meaning “high, exalted, lofty, sublime”.
Origin: uncertain; Arabic
Meaning: from Basque “joyful, happy”; “high, exalted, lofty, sublime”
Usage: Basque, English (as a variant spelling of Alia)
Maite is a Basque female name meaning “love, beloved, lovable”.
Maite has also been used as a Spanish nickname, or a contracted form of María Teresa. María is the Spanish, Galician, and Icelandic form of Mary which ultimately derives from Hebrew Miriam, the name of the sister of Moses and Aaron in the Old Testament. Miriam is a name of uncertain origin though several meanings have been ascribed to it such as “sea of bitterness”, “rebelliousness”, “obstinacy” or “wished for child” from a Hebrew root word. It’s also possible that it might have originated from an Egyptian name either meaning “beloved” from myr or “love” via mr (this latter explanation might be more likely since the names of Moses and Aaron have also been linked to a possible Egyptian origin).
Teresa is also a name of uncertain origin, the Spanish and Italian form of Theresa. As a given name it was first recorded in Spain, first used by the Spanish wife of St. Paulinus of Nola (a Roman poet, writer, and senator), though she was known as Therasia of Nola- her name seems to have derived from the name of a Greek island called Therasia, also known as Thirasía; it may be derived from Ancient Greek théros θέρος“summer; heat; harvest, crop” or from therízo θερίζω “to reap, cut”. I’ve also seen it linked to Ancient Greek thḗr θήρ “wild beast; beast of prey” or thēráō θηράω “to hunt”, but as intriguing as that meaning is I’m not too sold on it actually being related to the name.
It’s also possible that the name may have originated from a non PIE source.
Origin: Basque; uncertain
Meaning: from Basque “love, beloved, lovable”; also a contracted form of Spanish María Teresa