Each clue has an associated word, use the associated words to give you the final answer.
Clue #1: Head, Green, Broken
Clue #2: Neck, Twist, Fighter
Clue #3: Theory, Quartet, Cheese
Positive this was already basically answered by @marius and will delete if that answer gets undeleted but:
Clue #1: Head, Green, Broken
Arrow (arrow head, green arrow, broken arrow)
Clue #2: Neck, Twist, Fighter
Tie/Thai (neck tie, twist tie, thai fighter)
Clue #3: Theory, Quartet, Cheese
String (string theory, string quartet, cheese string/string cheese)
Title: Came up with this in the bar room meeting
Staff (bar staff, staff room, staff meeting)
Final Answer
Bow/Bō (bow and arrow, bow tie, bow string, Bō staff)
Clue #1: Head, Green, Broken
Arrow
Clue #2: Neck, Twist, Fighter
Bow
Clue #3: Theory, Quartet, Cheese
String
Which gives us the answer:
Archery