This case has finally given up a thread to pull on.
Tips
memory
线 (thread) + 头 (head/end). 头 here means 'tip/end' — the same use as in 街头 (street end, i.e. the corner) or 火车头 (locomotive, lit. 'train-head'). So 线头 = 'thread tip', whether the loose end on a shirt or a metaphorical lead in an investigation.
usage
Three concrete senses worth distinguishing: (1) the literal end of a length of thread you're sewing with; (2) an unwanted loose thread on a finished garment (你身上有线头); (3) figurative — a 'thread/lead' to follow in a mystery or problem.
