For alot of peeps north of 40 this is very, very normal.
Morning stiffness (!) is generally due to a very low grade inflammatory process that's ongoing due to your weekly milage or use. Your joints are in decline, albeit very, very slowly.
During the day our resting blood pressure keeps the amount of inflammatory exudate/grot in check by constanly moving it on. When you sleep, your blood prerssure drops and becomes labile and so the amount of inflammatory grot pools and stay in one place.
When you wake up your BP rises, when you get up the rise in BP increases because your head is now higher than your heart - the pooling inflammatory grot is now on the move and you feel the residual stiffness.
Stop running and it will stop....don't stop running, wear great shoes and modify what you do and it'll all be dandy. It's not overtraining but it's under recovery...