Cameron familiarly brought up European dodge (5)
I believe the answer is:
evade
'dodge' is the definition.
('evade' can be a synonym of 'dodge')
'cameron familiarly brought up european' is the wordplay.
'cameron familiarly' becomes 'dave' (David Cameron).
'brought up' is a reversal indicator (down clues only).
'european' becomes 'e' (abbreviation e.g. EU).
'dave' in reverse letter order is 'evad'.
'evad'+'e'='EVADE'
(Other definitions for evade that I've seen before include "Artfully dodge" , "Dodge, stay clear of" , "Dodge, keep clear of" , "Dodge - prevaricate" , "skirt" .)