Stone means nothing to a pet dog (4)
I believe the answer is:
opal
'stone' is the definition.
(I've seen this before)
'nothing to a pet dog' is the wordplay.
'nothing' becomes 'O' (looks like zero - 0).
'to a pet dog' becomes 'pal' (I can't explain this - if you can you should give a lot more credence to this answer).
'o'+'pal'='OPAL'
'means' is the link.
Can you help me to learn more?
(Other definitions for opal that I've seen before include "Glassy gemstone" , "Translucent gem" , "Lustrous stone" , "One's stony" , "Gem stone" .)