Eccentric names minister's house (5)
I believe the answer is:
manse
'minister's house' is the definition.
(house assigned to a religious minister)
'eccentric names' is the wordplay.
'eccentric' indicates an anagram.
'names' anagrammed gives 'MANSE'.
(Other definitions for manse that I've seen before include "Scottish minister's house" , "Presbyterian minister's residence" , "House of a clergyman" , "Names (anag.)" , "Scottish minister's residence" .)