Performance advances have made software packet processing a compelling alternative to hardware. However, software still lacks the delay predictability of hardware, an important property for security and quality of service. As a solution, we introduce PathMiner, an analysis tool that automatically builds performance models for software packet processors at the fine granularity of per-packet execution times. PathMiner combines symbolic execution and genetic algorithms in an iterative feedback loop to rapidly mine a packet processor binary for diverse packets that invoke complex execution paths. With these packets, PathMiner trains machine learning models that predict packet execution times and paths based on raw packet header bytes. We implement a prototype of PathMiner and test it by profiling a software IP router. The evaluation shows that PathMiner's models predict delay with low error-for over 40% of packets they predicted the router's execution time to within 10 cycles. Closer examination shows that PathMiner's effectiveness is due to higher execution path coverage than symbolic execution and the capability to generate diverse training samples efficiently. PathMiner takes an important step towards making it practical to predict delay for any software packet processor.