Check for codes.
Because of the cyclic nature of the priming, it is likely NOT the inertia switch, FP relay (itself), etc. It could be a puter issue (sending a ground pulse to the FP relay repeatedly), a bad ignition switch connection (the puter keeps thinking you're turning the key on), etc. You should be spitting a primary or secondary fuel circuit code with this issue. That could help narrow it down.
The underlying theory: when you turn the key on, the pump primes for 2-5 seconds. It remains off until the PIP circuit is active, at which time the pump runs non-stop.