In this work, a predictive eco-driving assistance system (pEDAS) with the goal to assist drivers in improving their driving style and thereby reducing the energy consumption in battery electric vehicles (BEVs) while enhancing the driving safety and comfort is introduced and evaluated. pEDAS in this work is equipped with two model predictive controllers (MPCs), namely reference-tracking MPC and car-following MPC, that use the information from onboard sensors, signal phase and timing (SPaT) messages from traffic light infrastructure, and geographical information of the driving route to compute an energy-optimal driving speed. An optimal speed suggestion and informative advice are indicated to the driver using a visual feedback. Moreover, the warning alerts during unsafe car-following situations are provided through an auditory feedback. pEDAS provides continuous feedback and encourages the drivers to perform energy-efficient car-following while tracking a preceding vehicle, travel at safe speeds at turns and curved roads, drive at energy-optimal speed determined using dynamic programming in freeway scenarios, and travel with a green-wave optimal speed to cross the signalized intersections at a green phase whenever possible. Furthermore, to evaluate the efficacy of the proposed eco-driving assistance system, user studies were conducted with 41 participants on a dynamic driving simulator. The objective analysis revealed that the drivers achieved mean energy savings up to 10%, reduced the speed limit violations, and avoided unnecessary stops at signalized intersections by using pEDAS. Finally, the user acceptance of the proposed pEDAS was evaluated using the Technology Acceptance Model (TAM) and Theory of Planned Behavior (TPB). The results showed an overall positive attitude of users and that the perceived usefulness and perceived behavioral control were found to be the significant factors in influencing the behavioral intention to use pEDAS.