One thinks there's wet weather at Birkenhead (5)

I believe the answer is:
brain
'one thinks there's wet' is the definition.
I can't tell whether this definition defines the answer.
'weather at birkenhead' is the wordplay.
'weather' becomes 'rain' (rain is a kind of weather).
'at' says to put letters next to each other (I've seen this in other clues).
'birkenhead' becomes 'b' (head letter of 'birken').
'rain' put after 'b' is 'BRAIN'.
Can you help me to learn more?
(Other definitions for brain that I've seen before include "It controls the central nervous system" , "Organ of the intellect" , "Thinking thing" , "Clever type" , "Clever person" .)
