Diagnosis
That stiffness is usually caused by mineral scale (hard water deposits) or grit building up in the diverter valve mechanism inside the filter head. It's a mechanical slide valve that gets gummed up.
How to Fix It
Remove the filter cartridge. Soak the entire filter head (the part that stays on your faucet) in a bowl of white vinegar for 30-60 minutes. The vinegar will dissolve the lime and calcium scale. After soaking, work the switch lever back and forth vigorously under running warm water to rinse it out and loosen the mechanism.
If That Didn't Work
The internal diverter valve might have a worn or broken plastic part. You can sometimes find replacement diverter valves (PUR part # varies) online. Otherwise, the whole filter head unit may need to be replaced.
Heads Up
Don't force the lever when it's stiff. You can snap the plastic internal linkage. Regular vinegar soaks every few months if you have hard water will keep it moving smoothly.