Bananas mean a lot to mum (3)
I believe the answer is:
mad
'bananas' is the definition.
(slang for insane)
'a lot to mum' is the wordplay.
'a lot' becomes 'd' (**).
'to' means one lot of letters go next to another (I've seen this in other clues).
'mum' becomes 'ma' ('ma' can be a synonym of 'mum').
'd' put after 'ma' is 'MAD'.
'mean' acts as a link.
(Other definitions for mad that I've seen before include "Demented or excited or angry" , "Angry or insane" , "Roused to anger" , "Frantic" , "Crazy" .)