By harbouring the sick, make a name (5)
I believe the answer is:
billy
'a name' is the definition.
'by harbouring the sick' is the wordplay.
'harbouring' is an insertion indicator.
'the sick' becomes 'ill' (I've seen this before).
'by' going around 'ill' is 'BILLY'.
'make' acts as a link.
(Other definitions for billy that I've seen before include "The familiar male goat" , "Light metal cooking pot used for camping" , "Camp can" , "Goat; camper's kettle" , "Cooking can for William" .)