Eccentric-(Adjective)
Definition:deviating from the recognized or customary character, practice, etc.; irregular; erratic; peculiar; odd.
Dictionary.com
The working basis of the establishment was an eccentric woman named
Caswell, a sort of musical enthusiast, who had a summer school of music
and the “higher things.” Malicious persons, not able to obtain
accommodations there, called the place “High C.”
The way the word is used in the passage is just suggesting that the women has a weird personality.
