Name borne by quiet anonymous Irish river (7)
I believe the answer is:
shannon
'irish river' is the definition.
(I've seen this before)
'name borne by quiet anonymous' is the wordplay.
'name' becomes 'n' (common abbreviation e.g. on forms).
'borne by' indicates putting letters inside (to bear can mean to carry).
'quiet' becomes 'sh'.
'anonymous' becomes 'anon'.
'sh'+'anon'='shanon'
'n' going inside 'shanon' is 'SHANNON'.
(Other definitions for shannon that I've seen before include "Irish banker" , "current in Ireland" , "River in Irish Republic" , "One who rendered runaway" , "Long-distance runner" .)