Me tipsy with gin: a mystery (6)
I believe the answer is:
enigma
'mystery' is the definition.
(synonyms)
'me tipsy with gin a' is the wordplay.
'tipsy with' is an anagram indicator.
'me' mixed up with 'gin' makes 'enigm'.
'enigm'+'a'='ENIGMA'
(Other definitions for enigma that I've seen before include "Difficult problem" , "It's mysterious like gamine" , "German military cipher machine" , "Riddle - German coding machine" , "It's mysterious, difficult to understand" .)