Mystery solved.

According to Wikipedia:

The Lake Forest Academy mascot is the "Caxy," which is ancient Greek for "ribbet" -- the croaking sound made by a frog. In the early 1900s, Aristophanes' hit comedy, The Frogs, was the subject of a popular Greek literature class.

I guess that explains where LFA got its nickname. But it doesn't explain why they chose it.

LFA is believed to be the only school with "Caxys" as a nickname...

Can't imagine why.