Other half grieve for Land of the Free (6)
I believe the answer is:
missus
'other half' is the definition.
(other half can mean one's spouse)
'grieve for land of the free' is the wordplay.
'grieve for' becomes 'miss' (to grieve for someone is to miss them).
'land of the free' becomes 'US' (United States of America).
'miss'+'us'='MISSUS'
(Other definitions for missus that I've seen before include "Dutch American" , "trouble and strife" , "The old lady" , "As partner" , "Wife (colloquial)" .)