Relative's favourite's raised accordingly by name (7)
I believe the answer is:
stepson
'relative's' is the definition.
(stepson is a kind of relative)
'favourite's raised accordingly by name' is the wordplay.
'favourite's' becomes 'pets' ('pet' can be a synonym of 'favourite').
'raised' is a reversal indicator.
'accordingly' becomes 'so' ('so' can be a synonym of 'accordingly').
'by' is a charade indicator (letters next to each other).
'name' becomes 'n' (common abbreviation e.g. on forms).
'pets' in reverse letter order is 'step'.
'step'+'so'+'n'='STEPSON'
(Other definitions for stepson that I've seen before include "Child from a spouse's former marriage" , "partner's sprog" , "No pests with relative by marriage" , "No pests about this relation by marriage" , "Child of a spouse by a former marriage" .)