Saw dog's name inside lead (7)
I believe the answer is:
proverb
'saw' is the definition.
(a saw can mean a proverb or maxim)
'dog's name inside lead' is the wordplay.
'dog's name' becomes 'rover' (I've seen this before).
'inside' means one lot of letters goes inside another.
'lead' becomes 'Pb' (Pb is the chemical symbol for lead).
'rover' going into 'pb' is 'PROVERB'.
(Other definitions for proverb that I've seen before include "Generally true saying" , "Familiar saying" , "Make haste slowly, perhaps" , "'Adage, saw (7)'" , "Old saw" .)