Thank you. Ayda is a beautiful common name in Iran. And about the meaning as Iranian names' website http://NameFarsi.com/نام-دختر says Aida (Ayda) آیدا has Azari origin which means inside of the moon, and in Persian it means happy as Italian.
Aida is feminine in Arabic. It means "to return" and Odeh is the male the version of Aida and it means "the return". Gift is "hadiyyah" and in older Arabic it is "Aaidyah".
Probably the meaning of "Aida" is connected with the noun "help", because each of the words "aid", "aide", "ayuda" means "help" as a noun, respectively in English, French, Spanish.
It says, under Ayda, that it means 'returning'. There's no reason it should have anything to do with the words for help in Romance languages (and 'aid' was borrowed from French into English), because it is an Arabic name.I love the sound of this ("ah-EE-da"), but I would absolutely cringe if it got pronounced "Ada" or "Ida". There is something dramatic, musical and womanly about the sound.