Relative, American, coming into money (6)
I believe the answer is:
cousin
'relative' is the definition.
(cousin is a kind of relative)
'american coming into money' is the wordplay.
'american' becomes 'US' (United States).
'coming into' indicates putting letters inside.
'money' becomes 'coin' (coin is a type of money).
'us' put inside 'coin' is 'COUSIN'.
(Other definitions for cousin that I've seen before include "Family relative" , "An aunt's or uncle's child" , "Aunt's/uncle's child" , "US icon is an odd relative" , "Relation, could be kissing" .)