Former prime minister's on board with cases (7)
I believe the answer is:
sheaths
'cases' is the definition.
('sheath' can be a synonym of 'case')
'former prime minister's on board' is the wordplay.
'former' becomes 'ss' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'prime minister' becomes 'Heath' (Edward Heath).
'on board' is an insertion indicator (inserted letters are taken on board).
'ss' going around 'heath' is 'SHEATHS'.
'with' acts as a link.
Can you help me to learn more?
(Other definitions for sheaths that I've seen before include "Protective covers" , "Cases" .)