By sheltering the sick, make a name (5)
I believe the answer is:
billy
'a name' is the definition.
'by sheltering the sick' is the wordplay.
'sheltering' means one lot of letters goes inside another.
'the sick' becomes 'ill' (I've seen this before).
'by' enclosing 'ill' is 'BILLY'.
'make' is the link.
(Other definitions for billy that I've seen before include "Metal camping can for tea" , "Goat; camper's kettle" , "... boy" , "Camper's pot" , "Cooking can for William" .)