Corrin may be a variation of
Corinne which is, in turn, the English and French variant of Corinna. It derives itself from the Greek word κόρη (kore), meaning "maiden." In Greek Mythology, this is the epithet of Persephone, who was kidnapped from her mother, Demeter, by Hades to become his wife. This could be reflecting the Avatar's own kidnapping from their family at the hands of King Garon.
Corinna is also the name of a Greek Lyric Poet from the 5th Century BC. It fell into common usage following the 17th century poem "Corinna's going a-Maying" by
Robert Herrick , whilst Corrine did not become popular until
Madame de Staël's 19th century novel of the same name.
Corrin may also be a variation of Corin which, while it can retain the same meaning as its counterpart, is the French variant of
Quirinus. Quirinus was the Sabine god of war. This could be a reference to the conflict between Hoshido and Nohr in which the Avatar is caught in the middle of. Quirinus is also derived from the Sabine word quirus, which means "spear."
While the name is presumably unisex, it seems to be more common as a feminine name.