Adaptation to various scene configurations and object properties, stability and dexterity in robotic grasping manipulation is far from explored. This work presents an origami-based shape morphing fingertip design to actively tackle the grasping stability and dexterity problems. The proposed fingertip utilizes origami as its skeleton providing degrees of freedom at desired positions and motor-driven four-bar-linkages as its transmission components to achieve a compact size of the fingertip. 3 morphing types that are commonly observed and essential in robotic grasping are studied and validated with geometrical modeling. Experiments including grasping an object with convex point contact to pivot or do pinch grasping, grasped object reorientation, and enveloping grasping with concave fingertip surfaces are implemented to demonstrate the advantages of our fingertip compared to conventional parallel grippers. Multi-functionality on enhancing grasping stability and dexterity via active adaptation given different grasped objects and manipulation tasks are justified. Video is available at youtu.be/jJoJ3xnDdVk/.