Put your name down (4)
I believe the answer is:
sign
'put' is the definition.
(signing is a kind of putting)
'your name down' is the wordplay.
I cannot really see how this works, but
'name' could be 'si' and 'si' is found within the answer.
This may be the basis of the clue (or it may be nonsense).
Can you help me to learn more?
(Other definitions for sign that I've seen before include "Public notice; Leo, say" , "Public notice; portent" , "Signal, gesture" , "Communicate silently" , "Autograph" .)