French Meaning: The name Avery is a French baby name. In French the meaning of the name Avery is: Rules with elf-wisdom. American Meaning: The name Avery is an American baby name. In American the meaning of the name Avery is: Rules with elf-wisdom.n Anglo-Saxon the meaning of the name Avery is: Rules the elves. English Meaning: The name Avery is an English baby name. In English the meaning of the name Avery is: Counselor; sage; wise.