From Wikipedia
"It was named by Scottish squatter Archibald Yuille established the first settlement, his sheep run called Ballaarat in 1837

with the name derived from local Wathaurong Aboriginal words for the area, balla arat, thought to mean "resting place". The present spelling was officially adopted by the City of Ballarat in 1996."
It has been spelt Ballarat for as long as I can recall (born 1964).