Songbird Greek character found on roofing material (8)
I believe the answer is:
nuthatch
'songbird' is the definition.
(nuthatch is a type of songbird)
'greek character found on roofing material' is the wordplay.
'greek character' becomes 'nu' (letter from the Greek alphabet).
'found on' says to put letters next to each other.
'roofing material' becomes 'thatch' (thatch is a kind of roofing material).
'nu'+'thatch'='NUTHATCH'
(Other definitions for nuthatch that I've seen before include "Tree-climbing songbird" , "Songbird named from its penchant for a kind of fruit" , "Tree-climbing bird" , "Songbird feeding on insects and kernels" , "Singer" .)