What could cause my sump pump to cycle on and off rapidly?
I'm not a sump pump expert, but it sounds like perhaps your check valve isn't working properly. If it isn't closing fully then water in the pipe can drain back down into the sump when the pump stops. If enough water is able to drain back down then it would cause the pump to start again.

Assuming you actually have a check valve, it probably looks something like this:



The lower the check valve is installed, the less water will be able to drain back down.