Can machine automatically generate multiple distinct and natural hand grasps, given specific contact region of an object in 3D? This motivates us to consider a novel task of Region Controllable Hand Grasp Generation (RegionGrasp), as follows: given as input a 3D object, together with its specific surface area selected as the intended contact region, to generate a diverse set of plausible hand grasps of the object, where the thumb finger tip touches the object surface on the contact region. To address this task, RegionGrasp-CVAE is proposed, which consists of two main parts. First, to enable contact region-awareness, we propose ConditionNet as the condition encoder that includes in it a transformer-backboned object encoder, O-Enc; a pretraining strategy is adopted by O-Enc, where the point patches of object surface are randomly masked off and subsequently restored, to further capture surface geometric information of the object. Second, to realize interaction awareness, HOINet is introduced to encode hand-object interaction features by entangling high-level hand features with embedded object features through geometric-aware multi-head cross attention. Empirical evaluations demonstrate the effectiveness of our approach qualitatively and quantitatively where it is shown to compare favorably with respect to the state of the art methods.