Indeed, but it’s perhaps more likely that the opera popularised the name in the 19th century, than an obscure medieval name being revived.
In any case, it’s possible that we are dealing with two names of different origin that are spelled the same? A (pseudo?) Greek name and a Germanic (?) name in the line of
Adelaïs or Alaïs.
Oddly enough I cannot find any French sources about this
Anaïs de Brienne, and she is completely absent in the French Wikipedia where all the children in the house of Brienne are mentioned, with sources. In other French sources this woman who was seduced by
Frederick II is called “
Marguerite”.
http://w.genealogy.euweb.cz/brienne/brienne1.html
This lack of sources is unexpected if such a obscure medieval figure would be the inspiration for the use of this name in
France. Perhaps some lost book telling the story became popular around 1800?