Brother married Greek, finally keeping old name (4)
I believe the answer is:
monk
'brother' is the definition.
(brother can mean a monk)
'married greek finally keeping old name' is the wordplay.
'married' becomes 'm' (abbreviation, e.g. in genealogy).
'finally' says to take the final letters.
'keeping' is an insertion indicator (some letters kept inside).
'old' becomes 'o' (abbreviation as in OT - Old Testament).
'name' becomes 'n' (common abbreviation e.g. on forms).
The final letter of 'greek' is 'k'.
'm'+'k'='mk'
'o'+'n'='on'
'mk' going around 'on' is 'MONK'.
(Other definitions for monk that I've seen before include "One vowed" , "Trappist, for example" , "Friar" , "Tonsured person" , "Setter" .)