Ancient blubber and bone one processed (5)
I believe the answer is:
niobe
'ancient blubber' is the definition.
(figure of Greek myth famous for crying or blubbing)
'bone one processed' is the wordplay.
'one' becomes 'i' (Roman numeral).
'processed' is an anagram indicator.
'bone'+'i'='bonei'
'bonei' with letters rearranged gives 'NIOBE'.
'and' is the link.
(Other definitions for niobe that I've seen before include "Daughter of Tantalus whose children were all slain" , "Weeping daughter of Tantalus" , "Weeping woman" , "Sad female" , "In Greek myth, turned into stone as she wept" .)